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D3×#144366 #72230404

order3072
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C29

Aliases

direct productD3×#144366

Splittings

FittingC3×#144366 ⋊ C2
SylowC3 ⋊ C2×#144366

Subgroups

subgroupG/H
centerC22D3×#2021
commutator, cosocleC23×C3×D4C24
FrattiniC23×D4C23×D3
FittingC3×#144366C2
socleC22×C3C2×#2021
hypercenter#144366D3
hypocenterC3C2×#144366
LCSG ▷ C23×C3×D4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #144366

34782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×B(5,𝔽2)
maximal normal subgroup ofD3×B(5,𝔽2)
automorphism group ofC2×C4×Dic3, C2×#217

GAP

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

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