D₁₅₇₁₅ #44537581

order	31430
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₇₁₅
unambiguous semidirect product	C₅⋊D₃₁₄₃
unambiguous semidirect product	C₇⋊D₂₂₄₅
unambiguous semidirect product	C₄₄₉⋊D₃₅

Splittings

Fitting, socle, derived, Hall	C₅×C₇×C₄₄₉ ⋊ C₂
Hall	C₇×C₄₄₉ ⋊ D₅
Hall	C₅×C₄₄₉ ⋊ D₇
Sylow	C₄₄₉ ⋊ D₃₅
Hall	C₅×C₇ ⋊ D₄₄₉
Sylow	C₇ ⋊ D₂₂₄₅
Sylow	C₅ ⋊ D₃₁₄₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₇×C₄₄₉	C₂
LCS	G ▷ C₅×C₇×C₄₄₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 461
transitive degree	μ_t(G) = 15715

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

Polycyclic presentation:

2,5,7,449|b^a=b^4,c^a=c^6,d^a=d^448

Conjugacy classes & character table

7859 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₇₁₅
maximal normal subgroup of	C₂×D₁₅₇₁₅
maximal quotient of	Dic₁₅₇₁₅, C₂×D₁₅₇₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31430, 9);

D15715 #44537581