#27158160

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #2258

Subgroups

	subgroup	G/H
center, Frattini	C₈	C₂²×#87604
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	M₃₂×C₉₇	C₄
prime radical	#13095511	C₈
cosocle	#13095486	C₂³
hypercenter	M₃₂	#87604
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ M₃₂

106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 3 generators on 161 points:

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

Polycyclic presentation:

2^7,97|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^22,c^b=cg,h^b=h^96,h^d=h^96

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20934);