C₂×#13164558 #72252726

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#13164558

Splittings

Fitting	C₂×C₃²×H(5,𝔽₂) ⋊ D₃
derived	C₃²⋊#997 ⋊ C₂²

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	C₃⋊#1020
commutator, cosocle, hypocenter	C₃²⋊#997	C₂²
Frattini	C₂×C₃	C₂×C₃⋊#1020
Fitting	C₂×C₃²×H(5,𝔽₂)	D₃
LCS	G ▷ C₃²⋊#997
UCS	C₁ ◁ C₂²×C₃

1629 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 1629 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 19
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 3 generators on 19 points:

(1,2)(3,5)(4,9)(6,10)(7,8)(12,15,16)(14,18,19);(4,8)(5,6)(9,10)(12,14)(15,19)(16,18);(4,5,10)(6,8,9)(11,12,14)(13,16,18)(15,19,17)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂×#72220369

GAP

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

gap> G := Group((1,2)(3,5)(4,9)(6,10)(7,8)(12,15,16)(14,18,19), (4,8)(5,6)(9,10)(12,14)(15,19)(16,18), (4,5,10)(6,8,9)(11,12,14)(13,16,18)(15,19,17));