Finite Group Databasebeta

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D3×#10603919 #72229628

order3840
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productD3×#10603919

Splittings

FittingC3×C5×#99 ⋊ C2×C4
HallC3×C5 ⋊ C2×#1867
SylowC5 ⋊ D3×#1867
SylowC3 ⋊ C2×#10603919

Subgroups

subgroupG/H
centerC2D3×#65699
commutatorC23×C3×C5C23×C4
FrattiniC23C22×D3×AGL(1,𝔽5)
FittingC3×C5×#99C2×C4
socleC2×C3×C5C2×#385
prime radicalC2×D3×D4×D5C22
cosocleC23×C3×D5C24
hypercenter#99D3×AGL(1,𝔽5)
hypocenterC3×C5C2×#1867
LCSG ▷ C23×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C23 ◁ #99

3988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,𝔽5)×B(4,𝔽2)
maximal normal subgroup ofD3×AGL(1,𝔽5)×B(4,𝔽2)

GAP

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

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