Finite Group DatabasebetaFinite Group Databasebeta| order | 30656 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic7664 |
| Sylow | C479 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7664 |
| commutator, cosocle | C16×C479 | C22 |
| Frattini | C16 | C2×D479 |
| Fitting | C32×C479 | C2 |
| socle, prime radical | C2×C479 | D16 |
| hypercenter | C32 | D479 |
| hypocenter | C479 | Q64 |
| LCS | G ▷ C16×C479 ▷ C8×C479 ▷ C4×C479 ▷ C2×C479 ▷ C479 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 543 |
| transitive degree | μt(G) = 30656 |
A minimal faithful permutation representation — 2 generators on 543 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,543)(67,542)(68,541)(69,540)(70,539)(71,538)(72,537)(73,536)(74,535)(75,534)(76,533)(77,532)(78,531)(79,530)(80,529)(81,528)(82,527)(83,526)(84,525)(85,524)(86,523)(87,522)(88,521)(89,520)(90,519)(91,518)(92,517)(93,516)(94,515)(95,514)(96,513)(97,512)(98,511)(99,510)(100,509)(101,508)(102,507)(103,506)(104,505)(105,504)(106,503)(107,502)(108,501)(109,500)(110,499)(111,498)(112,497)(113,496)(114,495)(115,494)(116,493)(117,492)(118,491)(119,490)(120,489)(121,488)(122,487)(123,486)(124,485)(125,484)(126,483)(127,482)(128,481)(129,480)(130,479)(131,478)(132,477)(133,476)(134,475)(135,474)(136,473)(137,472)(138,471)(139,470)(140,469)(141,468)(142,467)(143,466)(144,465)(145,464)(146,463)(147,462)(148,461)(149,460)(150,459)(151,458)(152,457)(153,456)(154,455)(155,454)(156,453)(157,452)(158,451)(159,450)(160,449)(161,448)(162,447)(163,446)(164,445)(165,444)(166,443)(167,442)(168,441)(169,440)(170,439)(171,438)(172,437)(173,436)(174,435)(175,434)(176,433)(177,432)(178,431)(179,430)(180,429)(181,428)(182,427)(183,426)(184,425)(185,424)(186,423)(187,422)(188,421)(189,420)(190,419)(191,418)(192,417)(193,416)(194,415)(195,414)(196,413)(197,412)(198,411)(199,410)(200,409)(201,408)(202,407)(203,406)(204,405)(205,404)(206,403)(207,402)(208,401)(209,400)(210,399)(211,398)(212,397)(213,396)(214,395)(215,394)(216,393)(217,392)(218,391)(219,390)(220,389)(221,388)(222,387)(223,386)(224,385)(225,384)(226,383)(227,382)(228,381)(229,380)(230,379)(231,378)(232,377)(233,376)(234,375)(235,374)(236,373)(237,372)(238,371)(239,370)(240,369)(241,368)(242,367)(243,366)(244,365)(245,364)(246,363)(247,362)(248,361)(249,360)(250,359)(251,358)(252,357)(253,356)(254,355)(255,354)(256,353)(257,352)(258,351)(259,350)(260,349)(261,348)(262,347)(263,346)(264,345)(265,344)(266,343)(267,342)(268,341)(269,340)(270,339)(271,338)(272,337)(273,336)(274,335)(275,334)(276,333)(277,332)(278,331)(279,330)(280,329)(281,328)(282,327)(283,326)(284,325)(285,324)(286,323)(287,322)(288,321)(289,320)(290,319)(291,318)(292,317)(293,316)(294,315)(295,314)(296,313)(297,312)(298,311)(299,310)(300,309)(301,308)(302,307)(303,306)(304,305)Polycyclic presentation:
2^6,479|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^478,c^b=cd,d^b=de,e^b=ef,g^b=g^4787667 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30656, 390);