Finite Group Databasebeta

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C3×D5×#932 #72229613

order2880
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×D5×#932

Splittings

FittingC2×C32×C5×D4 ⋊ C22
HallC32×C5 ⋊ C2×AGL(1,ℤ/8)
SylowC32 ⋊ D5×AGL(1,ℤ/8)
SylowC5 ⋊ C2×C3×#932

Subgroups

subgroupG/H
centerC2×C3C2×D5×#67
commutator, cosocleC3×C4×C5C24×C3
FrattiniC4C22×C3×D3×D5
FittingC2×C32×C5×D4C22
socleC2×C32×C5C22×D4
hypercenterC2×C3×D4D3×D5
hypocenterC3×C5C2×C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 120

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

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

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×#10595100, C3×AGL(1,𝔽5)×#932, #72230457
maximal normal subgroup ofD5×#10595100, C3×AGL(1,𝔽5)×#932, #72230457

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