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C13×#239 #10597784

order624
automorphism?
homologyAbelianization (H1) = C23×C13 · Schur multiplier (H2) = C22

Aliases

direct productC13×#239

Splittings

FittingC3×Q8×C13 ⋊ C2
HallC3×C13 ⋊ #41
SylowC3 ⋊ C13×#41

Subgroups

subgroupG/H
centerC2×C13C22×D3
commutator, cosocleC2×C3C23×C13
FrattiniC2C22×D3×C13
FittingC3×Q8×C13C2
socleC2×C3×C13C23
hypercenterQ8×C13D3
hypocenterC3C13×#41
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C13 ◁ Q8×C13

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:

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

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11928645, #11928651, #11928658, …
maximal normal subgroup of#11928645, #11928651, #11928658, …
maximal quotient ofC13×#890, C13×#892, C13×#893, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 221);