C₂×#11714169 #72268278

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

Aliases

direct product

C₂×#11714169

Splittings

Fitting

C₂×#3025 ⋊ D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#1020⋊C₂
commutator, cosocle, hypocenter	#997	C₂⁴
Frattini	C₂³	C₂³×S₄
Fitting	C₂×#3025	D₃
socle	C₂²	C₂×#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₂×C₄

3289 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 3289 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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72233471, C₂×#72236922, C₂×#72236838
maximal normal subgroup of	C₂×#72233471, C₂×#72236922, C₂×#72236838

GAP

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

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