Dic₁₃₆₅₆ #72289830

order	54624
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₆₅₆

Splittings

Hall	C₃×C₅₆₉ ⋊ Q₃₂
Sylow	C₅₆₉ ⋊ Dic₂₄
Sylow	C₃ ⋊ Dic₄₅₅₂

Subgroups

	subgroup	G/H
center	C₂	D₁₃₆₅₆
commutator, cosocle	C₃×C₈×C₅₆₉	C₂²
Frattini	C₈	C₂×D₁₇₀₇
Fitting	C₃×C₁₆×C₅₆₉	C₂
socle, prime radical	C₂×C₃×C₅₆₉	D₈
hypercenter	C₁₆	D₁₇₀₇
hypocenter	C₃×C₅₆₉	Q₃₂
LCS	G ▷ C₃×C₈×C₅₆₉ ▷ C₃×C₄×C₅₆₉ ▷ C₂×C₃×C₅₆₉ ▷ C₃×C₅₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 604
transitive degree	μ_t(G) = 54624

A minimal faithful permutation representation — 2 generators on 604 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,604)(38,603)(39,602)(40,601)(41,600)(42,599)(43,598)(44,597)(45,596)(46,595)(47,594)(48,593)(49,592)(50,591)(51,590)(52,589)(53,588)(54,587)(55,586)(56,585)(57,584)(58,583)(59,582)(60,581)(61,580)(62,579)(63,578)(64,577)(65,576)(66,575)(67,574)(68,573)(69,572)(70,571)(71,570)(72,569)(73,568)(74,567)(75,566)(76,565)(77,564)(78,563)(79,562)(80,561)(81,560)(82,559)(83,558)(84,557)(85,556)(86,555)(87,554)(88,553)(89,552)(90,551)(91,550)(92,549)(93,548)(94,547)(95,546)(96,545)(97,544)(98,543)(99,542)(100,541)(101,540)(102,539)(103,538)(104,537)(105,536)(106,535)(107,534)(108,533)(109,532)(110,531)(111,530)(112,529)(113,528)(114,527)(115,526)(116,525)(117,524)(118,523)(119,522)(120,521)(121,520)(122,519)(123,518)(124,517)(125,516)(126,515)(127,514)(128,513)(129,512)(130,511)(131,510)(132,509)(133,508)(134,507)(135,506)(136,505)(137,504)(138,503)(139,502)(140,501)(141,500)(142,499)(143,498)(144,497)(145,496)(146,495)(147,494)(148,493)(149,492)(150,491)(151,490)(152,489)(153,488)(154,487)(155,486)(156,485)(157,484)(158,483)(159,482)(160,481)(161,480)(162,479)(163,478)(164,477)(165,476)(166,475)(167,474)(168,473)(169,472)(170,471)(171,470)(172,469)(173,468)(174,467)(175,466)(176,465)(177,464)(178,463)(179,462)(180,461)(181,460)(182,459)(183,458)(184,457)(185,456)(186,455)(187,454)(188,453)(189,452)(190,451)(191,450)(192,449)(193,448)(194,447)(195,446)(196,445)(197,444)(198,443)(199,442)(200,441)(201,440)(202,439)(203,438)(204,437)(205,436)(206,435)(207,434)(208,433)(209,432)(210,431)(211,430)(212,429)(213,428)(214,427)(215,426)(216,425)(217,424)(218,423)(219,422)(220,421)(221,420)(222,419)(223,418)(224,417)(225,416)(226,415)(227,414)(228,413)(229,412)(230,411)(231,410)(232,409)(233,408)(234,407)(235,406)(236,405)(237,404)(238,403)(239,402)(240,401)(241,400)(242,399)(243,398)(244,397)(245,396)(246,395)(247,394)(248,393)(249,392)(250,391)(251,390)(252,389)(253,388)(254,387)(255,386)(256,385)(257,384)(258,383)(259,382)(260,381)(261,380)(262,379)(263,378)(264,377)(265,376)(266,375)(267,374)(268,373)(269,372)(270,371)(271,370)(272,369)(273,368)(274,367)(275,366)(276,365)(277,364)(278,363)(279,362)(280,361)(281,360)(282,359)(283,358)(284,357)(285,356)(286,355)(287,354)(288,353)(289,352)(290,351)(291,350)(292,349)(293,348)(294,347)(295,346)(296,345)(297,344)(298,343)(299,342)(300,341)(301,340)(302,339)(303,338)(304,337)(305,336)(306,335)(307,334)(308,333)(309,332)(310,331)(311,330)(312,329)(313,328)(314,327)(315,326)(316,325)(317,324)(318,323)(319,322)(320,321)

Polycyclic presentation:

2^5,3,569|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^568

Conjugacy classes & character table

13659 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 569]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^568);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13656 #72289830