#27166317

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

Polycyclic presentation:

2,3,7,307|d^a=d^306,c^b=c^2,d^b=d^17

Conjugacy classes & character table

367 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#27166317⋊C₂, C₂×#27166317
maximal normal subgroup of	#27166317⋊C₂, C₂×#27166317
maximal quotient of	#38788632, C₂×#27166317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12894, 16);