#27153232

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	#548×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ #3265

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	#548×C₉₇	C₂
socle	C₂²×C₉₇	C₂²×D₄
hypercenter	#548	D₉₇
hypocenter	C₉₇	#3265
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #548

546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 109
transitive degree	μ_t(G) = 3104

A minimal faithful permutation representation — 4 generators on 109 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)(103,107)(104,109)(106,108);(99,101);(102,103,105,108)(104,107,109,106);(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)(100,101)(103,106)(104,109)(107,108)

Polycyclic presentation:

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

Conjugacy classes & character table

1229 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 16006);