#27166316

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	#20264517 ⋊ C₂
Hall	C₇×D₃₀₇ ⋊ C₃
Hall	D₃₀₇ ⋊ ΓL(1,𝔽₈)
Sylow	C₃₀₇ ⋊ C₂×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃₀₇⋊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:

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

Polycyclic presentation:

2,3,7,307|d^a=d^306,c^b=c^2,d^b=d^289

Conjugacy classes & character table

367 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#27166316⋊C₂, C₂×#27166316
maximal normal subgroup of	#27166316⋊C₂, C₂×#27166316
maximal quotient of	#38788631, C₂×#27166316

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12894, 15);