Finite Group Databasebeta

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C3×#32891 #10651498

order768
nilpotency class3
isoclinismisoclinic class of #32888
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#32891

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×H(5,𝔽2)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

3940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270059, D3×#32891
maximal subgroup ofC3×#6628867, #72270059, D3×#32891
maximal normal subgroup ofC3×#6628867, #72270059, D3×#32891

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 25879);