Finite Group Databasebeta

← Prev (#72229629) Next (#72229631) →

C3×#11751944 #72229630

order2880
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productC3×#11751944

Splittings

HallC32×C5 ⋊ #440
SylowC32 ⋊ #65735
SylowC5 ⋊ C3×#5018

Subgroups

subgroupG/H
centerC2×C3AGL(1,𝔽5)×#67
commutatorC3×C4×C5C22×C3×C4
FrattiniC4C2×C3×D3×AGL(1,𝔽5)
FittingC32×C5×D4C2×C4
socleC2×C32×C5C4×D4
prime radicalC32×D4×D5C22
cosocleC3×C4×D5C23×C3
hypercenterC3×D4D3×AGL(1,𝔽5)
hypocenterC3×C5C3×#440
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72230434, C3×AGL(1,𝔽5)×#932, #72230460
maximal normal subgroup of#72230434, C3×AGL(1,𝔽5)×#932, #72230460

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