C₂×#72218858 #72244635

order	46080
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#72218858

Splittings

Fitting	C₂⁶ ⋊ S₆
derived	#72209614 ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72218858
commutator, cosocle, hypocenter	#72209614	C₂³
Frattini	C₂⁴	C₂²×S₆
Fitting	C₂⁶	S₆
socle	C₂⁵	C₂×S₆
LCS	G ▷ #72209614
UCS	C₁ ◁ C₂

At least 8571 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has at least 8571 conjugacy classes of subgroups — too many to draw. Only its 23 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 — 3 generators on 30 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

automorphism group of

#3573, C₃×#585

GAP

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

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