#27144031

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

Aliases

No aliases for this group.

Splittings

Fitting	M₆₄×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #2258

Subgroups

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₉₇
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	M₆₄×C₉₇	C₂
prime radical	#13095511	C₈
cosocle	C₁₆×C₉₇	C₂³
hypercenter	M₆₄	D₉₇
hypocenter	C₉₇	#2258
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₁₆ ◁ M₆₄

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

Conjugacy classes & character table

2000 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 6805);