Finite Group Databasebeta

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C3×#11073 #10629680

order768
nilpotency class3
isoclinismisoclinic class of #9103
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC3×#11073

Subgroups

subgroupG/H
center, socleC22×C3#386
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1026 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)(5,7)(8,9,12,15,13,17,11,16)(10,14,18,22,19,23,20,21);(4,5,6,7)(8,10)(9,14)(11,20)(12,18)(13,19)(15,22)(16,21)(17,23);(9,15)(11,12)(14,21)(16,17)(18,20)(22,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#11073, #72248749, #72248816, …
maximal subgroup ofD3×#11073, #72248749, C3×#417670, …
maximal normal subgroup ofD3×#11073, #72248749, C3×#417670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4061);