Dic₂₇₃₅ #24926476

order	10940
automorphism	Aut(G) order 11946480
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₂₇₃₅
unambiguous semidirect product	C₅₄₇⋊Dic₅

Splittings

derived, Hall	C₅×C₅₄₇ ⋊ C₄
Sylow	C₅₄₇ ⋊ Dic₅
Sylow	C₅ ⋊ Dic₅₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₂₇₃₅
commutator, hypocenter	C₅×C₅₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₅₄₇	C₂
LCS	G ▷ C₅×C₅₄₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 556
transitive degree	μ_t(G) = 10940

A minimal faithful permutation representation — 2 generators on 556 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556);(1,3,2,4)(6,9)(7,8)(11,556)(12,555)(13,554)(14,553)(15,552)(16,551)(17,550)(18,549)(19,548)(20,547)(21,546)(22,545)(23,544)(24,543)(25,542)(26,541)(27,540)(28,539)(29,538)(30,537)(31,536)(32,535)(33,534)(34,533)(35,532)(36,531)(37,530)(38,529)(39,528)(40,527)(41,526)(42,525)(43,524)(44,523)(45,522)(46,521)(47,520)(48,519)(49,518)(50,517)(51,516)(52,515)(53,514)(54,513)(55,512)(56,511)(57,510)(58,509)(59,508)(60,507)(61,506)(62,505)(63,504)(64,503)(65,502)(66,501)(67,500)(68,499)(69,498)(70,497)(71,496)(72,495)(73,494)(74,493)(75,492)(76,491)(77,490)(78,489)(79,488)(80,487)(81,486)(82,485)(83,484)(84,483)(85,482)(86,481)(87,480)(88,479)(89,478)(90,477)(91,476)(92,475)(93,474)(94,473)(95,472)(96,471)(97,470)(98,469)(99,468)(100,467)(101,466)(102,465)(103,464)(104,463)(105,462)(106,461)(107,460)(108,459)(109,458)(110,457)(111,456)(112,455)(113,454)(114,453)(115,452)(116,451)(117,450)(118,449)(119,448)(120,447)(121,446)(122,445)(123,444)(124,443)(125,442)(126,441)(127,440)(128,439)(129,438)(130,437)(131,436)(132,435)(133,434)(134,433)(135,432)(136,431)(137,430)(138,429)(139,428)(140,427)(141,426)(142,425)(143,424)(144,423)(145,422)(146,421)(147,420)(148,419)(149,418)(150,417)(151,416)(152,415)(153,414)(154,413)(155,412)(156,411)(157,410)(158,409)(159,408)(160,407)(161,406)(162,405)(163,404)(164,403)(165,402)(166,401)(167,400)(168,399)(169,398)(170,397)(171,396)(172,395)(173,394)(174,393)(175,392)(176,391)(177,390)(178,389)(179,388)(180,387)(181,386)(182,385)(183,384)(184,383)(185,382)(186,381)(187,380)(188,379)(189,378)(190,377)(191,376)(192,375)(193,374)(194,373)(195,372)(196,371)(197,370)(198,369)(199,368)(200,367)(201,366)(202,365)(203,364)(204,363)(205,362)(206,361)(207,360)(208,359)(209,358)(210,357)(211,356)(212,355)(213,354)(214,353)(215,352)(216,351)(217,350)(218,349)(219,348)(220,347)(221,346)(222,345)(223,344)(224,343)(225,342)(226,341)(227,340)(228,339)(229,338)(230,337)(231,336)(232,335)(233,334)(234,333)(235,332)(236,331)(237,330)(238,329)(239,328)(240,327)(241,326)(242,325)(243,324)(244,323)(245,322)(246,321)(247,320)(248,319)(249,318)(250,317)(251,316)(252,315)(253,314)(254,313)(255,312)(256,311)(257,310)(258,309)(259,308)(260,307)(261,306)(262,305)(263,304)(264,303)(265,302)(266,301)(267,300)(268,299)(269,298)(270,297)(271,296)(272,295)(273,294)(274,293)(275,292)(276,291)(277,290)(278,289)(279,288)(280,287)(281,286)(282,285)(283,284)

Polycyclic presentation:

2,5,547,2|a^2=d,b^a=b^4,c^a=c^546

Conjugacy classes & character table

2738 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×Dic₂₇₃₅, Dic₈₂₀₅, (C₅₄₇⋊C₃)⋊Dic₅, …
maximal normal subgroup of	C₃×Dic₂₇₃₅, (C₅₄₇⋊C₃)⋊Dic₅, Dic₅₄₇₀
maximal quotient of	C₃×Dic₂₇₃₅, Dic₈₂₀₅, Dic₁₃₆₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10940, 4);

Dic2735 #24926476