#27159033

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	C₂	C₂²×#13095525
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	Q₈×C₉₇	C₁₆
prime radical	#13095512	C₈
cosocle	#13095523	C₂³
hypercenter	Q₈	#13095525
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ Q₈

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)(3,24)(4,7)(5,34)(6,35)(8,11)(9,53)(10,54)(12,15)(13,16)(14,76)(17,89)(18,21)(19,86)(20,47)(22,26)(23,27)(25,39)(28,74)(29,32)(30,56)(31,51)(33,36)(37,78)(38,40)(41,75)(42,45)(43,46)(44,80)(48,57)(49,50)(52,59)(55,92)(58,64)(60,95)(61,63)(62,83)(65,94)(66,68)(69,72)(70,73)(71,84)(77,81)(79,90)(82,85)(87,97)(88,96)(91,93)(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)(3,8)(5,12)(6,13)(7,18)(9,22)(10,23)(11,29)(14,33)(15,38)(16,42)(17,46)(19,40)(20,49)(21,51)(24,56)(25,52)(26,61)(27,43)(28,66)(30,69)(31,70)(32,41)(34,78)(35,44)(36,62)(37,71)(39,75)(45,68)(47,85)(48,84)(50,83)(53,74)(54,77)(55,90)(57,91)(58,92)(59,94)(60,86)(63,95)(64,87)(65,88)(67,76)(72,73)(79,82)(80,81)(89,96)(93,97)(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);(2,3,5,9,6,10,14,25,4,8,12,22,13,23,33,52)(7,17,19,47,20,48,21,45,18,46,40,85,49,84,51,68)(11,28,30,67,31,38,71,75,29,66,69,76,70,15,37,39)(16,41,43,87,44,53,88,89,42,32,27,64,35,74,65,96)(24,55,57,73,58,63,93,59,56,90,91,72,92,95,97,94)(26,60,62,80,34,77,79,50,61,86,36,81,78,54,82,83)(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^8,c^b=cg,h^b=h^96,h^c=h^96,h^d=h^64,h^e=h^22,h^f=h^96

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21807);