#27166321

order	12894
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₃₀₇ ⋊ C₂×C₃
Hall	#20264516 ⋊ C₂
Hall	D₇×C₃₀₇ ⋊ C₃
Sylow	C₃₀₇ ⋊ AGL(1,𝔽₇)
Hall	D₇ ⋊ C₃₀₇⋊C₃
Sylow	C₇ ⋊ C₂×C₃₀₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₃₀₇	C₂×C₃
LCS	G ▷ C₇×C₃₀₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 314
transitive degree	μ_t(G) = 2149

A minimal faithful permutation representation — 2 generators on 314 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,209)(151,181,206)(154,184,203)(157,187,200)(160,190,197)(163,179,194)(166,195,192)(169,198,189)(172,201,186)(175,204,183)(177,207,180)(182,212,239)(185,215,236)(188,218,233)(191,221,230)(193,213,227)(196,226,225)(199,228,223)(202,231,220)(205,234,217)(208,237,214)(210,240,211)(216,243,264)(219,246,261)(222,249,258)(224,244,255)(229,254,253)(232,256,251)(235,259,248)(238,262,245)(241,265,242)(247,268,283)(250,271,280)(252,269,277)(257,276,275)(260,278,273)(263,281,270)(266,284,267)(272,287,296)(274,288,293)(279,292,291)(282,294,289)(285,297,286)(290,300,303)(295,302,301)(298,304,299)(305,307,306)(309,312,313,310,311,314);(1,2,4,8,15,25,38,54,73,95,120,148,179,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,213,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,216,244,190,128,78,40,14,23,34,47,63,82,104,129,157,188,219,247,269,221,156,100,56,24,36,50,66,85,107,132,160,191,222,250,272,288,249,187,125,75,37,52,69,88,110,135,163,193,224,252,274,290,300,271,218,153,97,53,71,91,113,138,166,196,209,239,264,283,296,303,287,246,184,122,72,93,116,141,169,199,229,240,176,206,236,261,280,293,268,215,150,94,118,144,172,202,232,257,265,207,145,173,203,233,258,277,243,181,119,146,175,205,235,260,279,284,237,174,117,142,170,200,230,255,212,147,177,208,238,263,282,295,297,262,204,143,92,114,139,167,197,227,178,210,241,266,285,298,305,304,281,234,171,115,70,89,111,136,164,194,225,253,275,291,301,306,307,294,259,201,140,90,51,67,86,108,133,161,192,223,251,273,289,299,302,278,231,168,112,68,35,48,64,83,105,130,158,189,220,248,270,286,292,256,198,137,87,49,22,32,45,61,80,102,127,155,186,217,245,267,276,228,165,109,65,33,12,19,29,42,58,77,99,124,152,183,214,242,254,195,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,211,226,162,106,62,30,10)(308,309,311,314,313,312,310)

Polycyclic presentation:

2,3,7,307|c^a=c^6,c^b=c^2,d^b=d^17

Conjugacy classes & character table

517 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#27166317⋊C₂, C₂×#27166321
maximal normal subgroup of	#27166317⋊C₂, C₂×#27166321
maximal quotient of	#38788636, C₂×#27166321

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12894, 20);