#34116533

order	20096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	Q₃₂×C₁₅₇ ⋊ C₄
Sylow	C₁₅₇ ⋊ #2176

Subgroups

	subgroup	G/H
center	C₂	D₈×#10597845
commutator	C₈×C₁₅₇	C₂²×C₄
Frattini	C₈	C₂²×#10597845
Fitting	Q₃₂×C₁₅₇	C₄
socle	C₂×C₁₅₇	C₄×D₈
prime radical	C₂×D₁₅₇	C₂×D₈
cosocle	C₈×D₁₅₇	C₂³
hypercenter	Q₃₂	#10597845
hypocenter	C₁₅₇	#2176
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 5024

A minimal faithful permutation representation — 3 generators on 189 points:

(1,2,8,4)(3,6,18,14)(5,7,16,10)(9,12,32,24)(11,17,28,20)(13,15,30,26)(19,22,50,38)(21,31,42,34)(23,25,48,44)(27,29,46,40)(33,36,72,56)(35,49,60,52)(37,39,70,62)(41,47,66,58)(43,45,68,64)(51,54,98,78)(53,71,82,74)(55,57,96,84)(59,69,88,80)(61,63,94,90)(65,67,92,86)(73,76,128,104)(75,97,108,100)(77,79,126,110)(81,95,114,106)(83,85,124,116)(87,93,120,112)(89,91,122,118)(99,102,149,134)(101,127,138,130)(103,105,148,139)(107,125,143,136)(109,111,131,144)(113,123,146,141)(115,117,137,147)(119,121,142,145)(129,132,140,151)(133,135,155,153)(150,152,154,157)(159,165)(164,179)(166,181)(167,177)(168,185)(172,184)(180,183)(182,188);(158,160,163,171)(159,164,165,179)(161,173,176,189)(162,178,174,170)(166,182,181,188)(167,183,177,180)(168,184,185,172)(169,186,187,175);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,83,84,70)(66,69,87,86)(67,89,90,68)(76,99,101,100)(79,103,104,98)(82,97,107,106)(85,109,110,96)(88,95,113,112)(91,115,116,94)(92,93,119,118)(102,129,131,130)(105,133,134,128)(108,127,137,136)(111,132,139,126)(114,125,142,141)(117,138,144,124)(120,123,146,145)(121,143,147,122)(135,150,151,149)(140,152,153,148)(154,156,157,155)(158,159)(160,168)(161,172)(162,177)(163,165)(164,169)(166,175)(167,174)(170,188)(171,185)(173,183)(176,184)(178,182)(179,187)(180,189)(181,186)

Polycyclic presentation:

2^7,157|a^2=e,b^2=g,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^28,d^b=df,f^b=fg,h^c=h^129,h^e=h^156

Conjugacy classes & character table

461 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20096, 20907);