C₂×#11715752 #72268258

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#11715752

Splittings

Fitting	C₂×#3529 ⋊ D₃
derived	#5465 ⋊ C₂³

Subgroups

	subgroup	G/H
center	C₂×C₄	#5934
commutator, cosocle, hypocenter	#5465	C₂³
Frattini	C₂²	C₂²×#1020
Fitting	C₂×#3529	D₃
socle	C₂²	C₂×#5934
hypercenter	C₂²×C₄	#1020
LCS	G ▷ #5465
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

2975 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 2975 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂×#72218487

GAP

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

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