Finite Group Databasebeta

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C3×#60359 #10678966

order768
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC3×#60359

Subgroups

subgroupG/H
centerC2×C3×C4C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
socleC22×C3C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C24×C3×C4 ◁ G

5098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253683, #72253686, D3×#60359, …
maximal subgroup ofC3×#6437445, #72253683, #72253686, …
maximal normal subgroup ofC3×#6437445, #72253683, #72253686, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 53347);