Finite Group Databasebeta

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C3×#6101 #10597632

order624
isoclinismisoclinic class of Dic26
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#6101

Splittings

HallC3×C13 ⋊ C4⋊C4
SylowC13 ⋊ C3×C4⋊C4

Subgroups

subgroupG/H
centerC22×C3C2×D13
commutatorC2×C13C2×C3×C4
FrattiniC22C2×C3×D13
FittingC2×C3×C4×C13C2
socle, prime radicalC22×C3×C13C22
cosocleC22×C13C22×C3
hypercenterC2×C3×C4D13
hypocenterC13C3×C4⋊C4
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 624

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11928425, #11928427, #11928428, …
maximal normal subgroup of#11928425, #11928427, #11928428, …
maximal quotient ofC3×#88047, C3×#88048, C3×#88054, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(624, 69);