C₂×#20261834 #27158911

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

Aliases

direct product

C₂×#20261834

Splittings

Sylow

C₉₇ ⋊ C₂×#349

Subgroups

	subgroup	G/H
center	C₂²	C₂²⋊#11715906
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₂²×C₄×C₉₇	C₈
socle	C₂²×C₉₇	C₂²⋊C₈
prime radical	C₂³×D₉₇	C₈
cosocle	C₂×#11715897	C₂³
hypercenter	C₂²×C₄	#11715906
hypocenter	C₉₇	C₂×#349
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

236 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 3104

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

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

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12416, 21685);