C₂×#21483⋊C₃ #72341695

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#21483⋊C₃

Splittings

Fitting, Sylow

C₂×#21483 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	#120⋊C₃
commutator, hypocenter	H(5,𝔽₂)	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂³×A₄
Fitting	C₂×#21483	C₃
socle	C₂²	C₄×#120⋊C₃
prime radical	C₂×#3025⋊C₃	C₂
cosocle	#585	C₂³×C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₈

1724 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 1724 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342027, C₂×#72236846, #72342051, …
maximal normal subgroup of	#72342027, C₂×#72236846, #72342051, …

GAP

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

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