Finite Group Databasebeta

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C3×#72224585 #72224614

order7200
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC3×#72224585

Splittings

FittingC22×C32×C52 ⋊ C2×C4
HallC32×C52 ⋊ C4×D4
SylowC52 ⋊ C3×C4×#67
SylowC32 ⋊ #11717375

Subgroups

subgroupG/H
centerC2×C3C2×#10597182
commutatorC2×C3×C52C22×C3×C4
FrattiniC2C2×C3×#10597182
FittingC22×C32×C52C2×C4
socleC2×C32×C52C22×C4
prime radicalC3×#11925664C2
cosocleC2×C3×C5⋊D5C23×C3
hypercenterC22×C3#10597182
hypocenterC3×C52C3×C4×D4
LCSG ▷ C2×C3×C52 ▷ C3×C52
UCSC1 ◁ C2×C3 ◁ C22×C3

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3)(4,5,6)(12,13,14,15)(16,17,18,19);(8,10)(11,12,15,13,14);(5,6)(7,8,9,10)(17,18)(19,20)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224650, C3×D3×#11717375, #72254735, …
maximal normal subgroup of#72224650, C3×D3×#11717375, C2×C3×#72224585
maximal quotient of#72254735, C2×C3×#72224585

GAP

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

gap> G := Group((1,2,3)(4,5,6)(12,13,14,15)(16,17,18,19), (8,10)(11,12,15,13,14), (5,6)(7,8,9,10)(17,18)(19,20));