C₂×#32896⋊C₃ #72342949

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#32896⋊C₃

Splittings

Fitting, Sylow

C₂×#32896 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	AΣL(1,𝔽₆₄)
commutator, cosocle	#585	C₂³×C₃
Frattini	C₂²×C₄	C₂³×A₄
Fitting	C₂×#32896	C₃
prime radical	C₂×#87547	C₂
hypercenter	C₂×D₄	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₂×C₃×D₄
LCS	G ▷ #585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

5902 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 5902 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236917, C₂×#72236912, C₂×#72236874, …
maximal normal subgroup of	C₂×#72236917, C₂×#72236912, C₂×#72236874, …

GAP

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

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