#27150258

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	#416×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #2897

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×D₉₇
commutator	C₄×C₉₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#416×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
prime radical	C₂×#15690856	C₂
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	#416	D₉₇
hypocenter	C₉₇	#2897
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 117 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)(104,111)(108,115)(112,117)(114,116);(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)(102,104)(103,108)(105,112)(106,111)(107,114)(109,116)(110,115)(113,117);(103,107)(105,113)(108,114)(109,110)(112,117)(115,116);(98,99,100,101)(102,103)(104,108)(105,107)(106,110)(109,113)(111,115)(112,114)(116,117)

Polycyclic presentation:

2^7,97|a^2=f,e^2=g,b^a=be,e^a=eg,e^b=eg,d^c=dg,h^c=h^96,h^d=h^96

Conjugacy classes & character table

1388 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 13032);