C₂×#72218848 #72244369

order	23040
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#72218848

Splittings

derived

#72209614 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72218848
commutator, cosocle, hypocenter	#72209614	C₂²
Frattini	C₂⁴	C₂×S₆
Fitting, socle	C₂⁵	S₆
LCS	G ▷ #72209614
UCS	C₁ ◁ C₂

2462 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 2462 conjugacy classes of subgroups — too many to draw. Only its 9 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) = 30
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72218858
maximal normal subgroup of	C₂×#72218858

GAP

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

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