Finite Group Databasebeta

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D3×#511790 #72231732

order3072
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C212

Aliases

direct productD3×#511790

Splittings

FittingC3×#511790 ⋊ C2
SylowC3 ⋊ C2×#511790

Subgroups

subgroupG/H
centerC2D3×#32898
commutator, cosocleC22×C3×D4C25
FrattiniC22×D4C24×D3
FittingC3×#511790C2
socleC2×C3C2×#32898
hypercenter#511790D3
hypocenterC3C2×#511790
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ #511790

At least 15996 conjugacy classes of subgroups; only the 1151 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#72209772, D3×B(5,𝔽2), D3×AGL(2,ℤ/4), …
maximal normal subgroup ofD3×B(5,𝔽2), C2×D3×#511790
maximal quotient ofD3×#72213329, C2×D3×#511790
automorphism group of#869, #4526
holomorph ofC3×C22⋊C4

GAP

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

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