Finite Group Databasebeta

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D3×#23895 #72231027

order1536
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C211

Aliases

direct productD3×#23895

Splittings

FittingC3×#23895 ⋊ C2
SylowC3 ⋊ C2×#23895

Subgroups

subgroupG/H
centerC2D3×#2403
commutator, cosocleC24×C3C25
FrattiniC24C24×D3
FittingC3×#23895C2
socleC2×C3C2×#2403
hypercenter#23895D3
hypocenterC3C2×#23895
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C24 ◁ #23895

14444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#6499874, D3×#511790, D3×#144378, …
maximal normal subgroup ofD3×#6499874, D3×#511790, D3×#144378, …
maximal quotient ofD3×#245225, D3×#394064, #72252356, …

GAP

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

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