Finite Group Databasebeta

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C3×#15314 #10633921

order768
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4

Aliases

direct productC3×#15314

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×#392C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72254660, #72254789, D3×#15314, …
maximal subgroup ofC3×#4312024, #72254660, #72254789, …
maximal normal subgroup ofC3×#4312024, #72254660, #72254789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8302);