C₂×#20261597 #27157279

order	12416
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#20261597

Splittings

Sylow

C₉₇ ⋊ C₂×#334

Subgroups

	subgroup	G/H
center	C₂³	C₄⋊#87604
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂²×C₈×C₉₇	C₄
socle, prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	C₂²×C₈	#87604
hypocenter	C₉₇	C₂×#334
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

200 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 12416

A minimal faithful permutation representation — 3 generators on 115 points:

(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,82)(48,49)(51,80)(52,53)(55,78)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,79)(69,70)(72,76)(73,74)(81,89)(83,88)(84,85)(86,87)(90,95)(91,92)(93,94)(96,97)(108,109,110,112,111,113,114,115);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(100,101,102,104,103,105,106,107)(109,112)(110,114)(113,115);(98,99)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 20053);