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C3×#6129 #10597772

order624
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#6129

Splittings

FittingC3×D4×C13 ⋊ C2
HallC3×C13 ⋊ #41
SylowC13 ⋊ C3×#41

Subgroups

subgroupG/H
centerC2×C3C22×D13
commutator, cosocleC2×C13C23×C3
FrattiniC2C22×C3×D13
FittingC3×D4×C13C2
socleC2×C3×C13C23
hypercenterC3×D4D13
hypocenterC13C3×#41
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C3 ◁ C3×D4

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 312

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3)(4,5,7,9,11,13,15,16,14,12,10,8,6)(18,22)(19,23);(17,19,20,23)(18,21,22,24);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#88116, #11928377, #11928619, …
maximal normal subgroup ofC3×#88116, #11928377, #11928619, …
maximal quotient ofC3×#88131, C3×#88132, C3×#88139, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 209);