C₂×#11715739 #72227313

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

Aliases

direct product

C₂×#11715739

Splittings

Fitting

C₂×#3573 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#1020
commutator, cosocle	#5950	C₂³
Frattini	C₄	C₂²×#1020
Fitting	C₂×#3573	D₃
hypercenter	C₂×C₈	#1020
hypocenter	#997	C₂×D₄
LCS	G ▷ #5950 ▷ #997
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

2867 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 2867 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72215718, #72342132, #72342144, …
maximal normal subgroup of	C₂×#72215718, #72342132, #72342144, …

GAP

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

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