C₂×#20260692 #27150150

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₂×#20260692

Splittings

Fitting	C₂×#97×C₉₇ ⋊ C₂
Sylow	C₉₇ ⋊ C₂×#405

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉₇
commutator	C₂×C₉₇	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#97×C₉₇	C₂
socle	C₂³×C₉₇	C₂²×C₄
prime radical	C₂³×D₉₇	C₂×C₄
cosocle	C₂×C₄×C₉₇	C₂⁴
hypercenter	C₂×#97	D₉₇
hypocenter	C₉₇	C₂×#405
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6208

A minimal faithful permutation representation — 4 generators on 119 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)(106,112)(109,115)(113,118)(116,119);(100,101,102,103)(104,106)(105,109)(107,113)(108,112)(110,116)(111,115)(114,118)(117,119);(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)(104,105,107,110,108,111,114,117)(106,109,113,116,112,115,118,119);(98,99)

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, 12924);

C2×#20260692 #27150150