#27139719

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Sylow

C₉₇ ⋊ #1338

Subgroups

	subgroup	G/H
center	C₂×C₄	#13095488
commutator	C₂×C₄×C₉₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₉₇
Fitting	#346×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₂²⋊C₈
cosocle	C₄×C₈×C₉₇	C₂²
hypercenter	#346	D₉₇
hypocenter	C₉₇	#1338
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 2 generators on 121 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)(98,99,101,103)(100,105,104,102)(106,108,110,111)(107,112,113,115)(109,117,118,116)(114,121,120,119);(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)(99,102)(100,104)(103,105)(106,107,109,114,111,112,117,119,110,113,118,120,108,115,116,121)

Polycyclic presentation:

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

Conjugacy classes & character table

1958 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 2493);