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C3×#10937 #10629544

order768
nilpotency class3
isoclinismisoclinic class of B(4,𝔽2)
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC3×#10937

Subgroups

subgroupG/H
centerC2×C3×C4#120
commutatorC23C2×C3×C42
Frattini, cosocleC23×C4C23×C3
socleC22×C3#386
prime radicalC25×C3C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248345, #72248346, #72248391, …
maximal subgroup ofC3×#421068, #72248345, #72248346, …
maximal normal subgroup ofC3×#421068, #72248345, #72248346, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3925);