C₂×D₃×Ω⁺(4,𝔽₃) #72242403

order	3456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂²×C₃

solvable

Aliases

direct product

C₂×D₃×Ω⁺(4,𝔽₃)

Splittings

Fitting	C₂×C₃×H(5,𝔽₂) ⋊ C₂×C₃²
derived	C₃×H(5,𝔽₂) ⋊ C₂²×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₃×A₄²
commutator, cosocle, hypocenter	C₃×H(5,𝔽₂)	C₂²×C₃²
Frattini	C₂	C₂×D₃×A₄²
Fitting	C₂×C₃×H(5,𝔽₂)	C₂×C₃²
socle	C₂²×C₃	C₂×A₄²
LCS	G ▷ C₃×H(5,𝔽₂)
UCS	C₁ ◁ C₂²

1568 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 91 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) = 29
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₃×SO⁺(4,𝔽₃), D₃×#983⋊A₄, C₂×#72220370
maximal normal subgroup of	C₂×D₃×SO⁺(4,𝔽₃), D₃×#983⋊A₄
maximal quotient of	C₂×#72220370

GAP

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

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