C₂×#11711643 #72341562

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#11711643

Splittings

Fitting

C₂×#1791 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	#72991
commutator, cosocle	#5950	C₂³
Frattini	C₂²×C₄	C₂²×S₄
Fitting	C₂×#1791	D₃
hypercenter	C₂×C₄	#1020⋊C₂
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1755 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 1755 conjugacy classes of subgroups — too many to draw. Only its 43 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:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236770, C₂×#72236763, C₂×#72236783, …
maximal normal subgroup of	C₂×#72236770, C₂×#72236763, C₂×#72236783, …

GAP

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

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