Dic₁₃₇₁₈ #72289929

order	54872
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₇₁₈

Splittings

Sylow

C₆₈₅₉ ⋊ Q₈

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₈₅₉
commutator, cosocle	C₂×C₆₈₅₉	C₂²
Frattini	C₂×C₃₆₁	C₂×D₁₉
Fitting	C₄×C₆₈₅₉	C₂
socle, prime radical	C₂×C₁₉	C₂×D₃₆₁
hypercenter	C₄	D₆₈₅₉
hypocenter	C₆₈₅₉	Q₈
LCS	G ▷ C₂×C₆₈₅₉ ▷ C₆₈₅₉
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 6867
transitive degree	μ_t(G) = 54872

A minimal faithful permutation representation — 2 generators on 6867 points:

(1,2,3,4)(5,8,7,6)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221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3883)(2995,3882)(2996,3881)(2997,3880)(2998,3879)(2999,3878)(3000,3877)(3001,3876)(3002,3875)(3003,3874)(3004,3873)(3005,3872)(3006,3871)(3007,3870)(3008,3869)(3009,3868)(3010,3867)(3011,3866)(3012,3865)(3013,3864)(3014,3863)(3015,3862)(3016,3861)(3017,3860)(3018,3859)(3019,3858)(3020,3857)(3021,3856)(3022,3855)(3023,3854)(3024,3853)(3025,3852)(3026,3851)(3027,3850)(3028,3849)(3029,3848)(3030,3847)(3031,3846)(3032,3845)(3033,3844)(3034,3843)(3035,3842)(3036,3841)(3037,3840)(3038,3839)(3039,3838)(3040,3837)(3041,3836)(3042,3835)(3043,3834)(3044,3833)(3045,3832)(3046,3831)(3047,3830)(3048,3829)(3049,3828)(3050,3827)(3051,3826)(3052,3825)(3053,3824)(3054,3823)(3055,3822)(3056,3821)(3057,3820)(3058,3819)(3059,3818)(3060,3817)(3061,3816)(3062,3815)(3063,3814)(3064,3813)(3065,3812)(3066,3811)(3067,3810)(3068,3809)(3069,3808)(3070,3807)(3071,3806)(3072,3805)(3073,3804)(3074,3803)(3075,3802)(3076,3801)(3077,3800)(3078,3799)(3079,3798)(3080,3797)(3081,3796)(3082,3795)(3083,3794)(3084,3793)(3085,3792)(3086,3791)(3087,3790)(3088,3789)(3089,3788)(3090,3787)(3091,3786)(3092,3785)(3093,3784)(3094,3783)(3095,3782)(3096,3781)(3097,3780)(3098,3779)(3099,3778)(3100,3777)(3101,3776)(3102,3775)(3103,3774)(3104,3773)(3105,3772)(3106,3771)(3107,3770)(3108,3769)(3109,3768)(3110,3767)(3111,3766)(3112,3765)(3113,3764)(3114,3763)(3115,3762)(3116,3761)(3117,3760)(3118,3759)(3119,3758)(3120,3757)(3121,3756)(3122,3755)(3123,3754)(3124,3753)(3125,3752)(3126,3751)(3127,3750)(3128,3749)(3129,3748)(3130,3747)(3131,3746)(3132,3745)(3133,3744)(3134,3743)(3135,3742)(3136,3741)(3137,3740)(3138,3739)(3139,3738)(3140,3737)(3141,3736)(3142,3735)(3143,3734)(3144,3733)(3145,3732)(3146,3731)(3147,3730)(3148,3729)(3149,3728)(3150,3727)(3151,3726)(3152,3725)(3153,3724)(3154,3723)(3155,3722)(3156,3721)(3157,3720)(3158,3719)(3159,3718)(3160,3717)(3161,3716)(3162,3715)(3163,3714)(3164,3713)(3165,3712)(3166,3711)(3167,3710)(3168,3709)(3169,3708)(3170,3707)(3171,3706)(3172,3705)(3173,3704)(3174,3703)(3175,3702)(3176,3701)(3177,3700)(3178,3699)(3179,3698)(3180,3697)(3181,3696)(3182,3695)(3183,3694)(3184,3693)(3185,3692)(3186,3691)(3187,3690)(3188,3689)(3189,3688)(3190,3687)(3191,3686)(3192,3685)(3193,3684)(3194,3683)(3195,3682)(3196,3681)(3197,3680)(3198,3679)(3199,3678)(3200,3677)(3201,3676)(3202,3675)(3203,3674)(3204,3673)(3205,3672)(3206,3671)(3207,3670)(3208,3669)(3209,3668)(3210,3667)(3211,3666)(3212,3665)(3213,3664)(3214,3663)(3215,3662)(3216,3661)(3217,3660)(3218,3659)(3219,3658)(3220,3657)(3221,3656)(3222,3655)(3223,3654)(3224,3653)(3225,3652)(3226,3651)(3227,3650)(3228,3649)(3229,3648)(3230,3647)(3231,3646)(3232,3645)(3233,3644)(3234,3643)(3235,3642)(3236,3641)(3237,3640)(3238,3639)(3239,3638)(3240,3637)(3241,3636)(3242,3635)(3243,3634)(3244,3633)(3245,3632)(3246,3631)(3247,3630)(3248,3629)(3249,3628)(3250,3627)(3251,3626)(3252,3625)(3253,3624)(3254,3623)(3255,3622)(3256,3621)(3257,3620)(3258,3619)(3259,3618)(3260,3617)(3261,3616)(3262,3615)(3263,3614)(3264,3613)(3265,3612)(3266,3611)(3267,3610)(3268,3609)(3269,3608)(3270,3607)(3271,3606)(3272,3605)(3273,3604)(3274,3603)(3275,3602)(3276,3601)(3277,3600)(3278,3599)(3279,3598)(3280,3597)(3281,3596)(3282,3595)(3283,3594)(3284,3593)(3285,3592)(3286,3591)(3287,3590)(3288,3589)(3289,3588)(3290,3587)(3291,3586)(3292,3585)(3293,3584)(3294,3583)(3295,3582)(3296,3581)(3297,3580)(3298,3579)(3299,3578)(3300,3577)(3301,3576)(3302,3575)(3303,3574)(3304,3573)(3305,3572)(3306,3571)(3307,3570)(3308,3569)(3309,3568)(3310,3567)(3311,3566)(3312,3565)(3313,3564)(3314,3563)(3315,3562)(3316,3561)(3317,3560)(3318,3559)(3319,3558)(3320,3557)(3321,3556)(3322,3555)(3323,3554)(3324,3553)(3325,3552)(3326,3551)(3327,3550)(3328,3549)(3329,3548)(3330,3547)(3331,3546)(3332,3545)(3333,3544)(3334,3543)(3335,3542)(3336,3541)(3337,3540)(3338,3539)(3339,3538)(3340,3537)(3341,3536)(3342,3535)(3343,3534)(3344,3533)(3345,3532)(3346,3531)(3347,3530)(3348,3529)(3349,3528)(3350,3527)(3351,3526)(3352,3525)(3353,3524)(3354,3523)(3355,3522)(3356,3521)(3357,3520)(3358,3519)(3359,3518)(3360,3517)(3361,3516)(3362,3515)(3363,3514)(3364,3513)(3365,3512)(3366,3511)(3367,3510)(3368,3509)(3369,3508)(3370,3507)(3371,3506)(3372,3505)(3373,3504)(3374,3503)(3375,3502)(3376,3501)(3377,3500)(3378,3499)(3379,3498)(3380,3497)(3381,3496)(3382,3495)(3383,3494)(3384,3493)(3385,3492)(3386,3491)(3387,3490)(3388,3489)(3389,3488)(3390,3487)(3391,3486)(3392,3485)(3393,3484)(3394,3483)(3395,3482)(3396,3481)(3397,3480)(3398,3479)(3399,3478)(3400,3477)(3401,3476)(3402,3475)(3403,3474)(3404,3473)(3405,3472)(3406,3471)(3407,3470)(3408,3469)(3409,3468)(3410,3467)(3411,3466)(3412,3465)(3413,3464)(3414,3463)(3415,3462)(3416,3461)(3417,3460)(3418,3459)(3419,3458)(3420,3457)(3421,3456)(3422,3455)(3423,3454)(3424,3453)(3425,3452)(3426,3451)(3427,3450)(3428,3449)(3429,3448)(3430,3447)(3431,3446)(3432,3445)(3433,3444)(3434,3443)(3435,3442)(3436,3441)(3437,3440)(3438,3439)

Polycyclic presentation:

2^3,19^3|a^2=c,b^2=c,d^19=e,e^19=f,b^a=bc,d^a=d^18e^18f^18,e^a=e^18f^18,f^a=f^18

Conjugacy classes & character table

13721 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 19, 19, 19]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^18*F.5^18*F.6^18);;
gap> SetConjugate(coll, 5, 1, F.5^18*F.6^18);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13718 #72289929