C₂×#11714172 #72343890

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#11714172

Splittings

Fitting

C₂²×#458 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂²×#458	D₃
hypercenter	C₂³	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂³

3300 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3300 conjugacy classes of subgroups — too many to draw. Only its 107 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

(3,15,6,31,8,17,19,29)(4,25,11,34,9,26,22,33)(5,16,14,18,13,7,20,30)(10,23,24,27,21,12,28,32);(4,9)(5,13)(6,16)(7,19)(11,12)(14,15)(17,20)(18,30)(22,23)(24,26)(25,28)(33,34);(3,4)(5,10)(6,12)(7,11)(8,9)(13,21)(14,25)(15,24)(16,22)(17,28)(18,27)(19,23)(20,26)(29,33)(30,32)(31,34);(1,2)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236841, C₂×#72236843, C₂×#72236838
maximal normal subgroup of	C₂×#72236841, C₂×#72236843, C₂×#72236838

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,15,6,31,8,17,19,29)(4,25,11,34,9,26,22,33)(5,16,14,18,13,7,20,30)(10,23,24,27,21,12,28,32), (4,9)(5,13)(6,16)(7,19)(11,12)(14,15)(17,20)(18,30)(22,23)(24,26)(25,28)(33,34), (3,4)(5,10)(6,12)(7,11)(8,9)(13,21)(14,25)(15,24)(16,22)(17,28)(18,27)(19,23)(20,26)(29,33)(30,32)(31,34), (1,2));