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C3×#10768 #10629375

order768
nilpotency class3
isoclinismisoclinic class of #1828
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26

Aliases

direct productC3×#10768

Subgroups

subgroupG/H
center, socleC22×C3#381
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

1498 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)(4,5)(6,7)(8,9)(10,13,17,21)(11,18,19,14)(12,15)(16,22)(20,23);(8,10)(9,13,15,21)(11,16)(12,17)(14,20,18,22)(19,23);(5,7)(8,11)(9,14)(10,16)(12,19)(13,20)(15,18)(17,23)(21,22)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270106, #72270155, D3×#10768
maximal subgroup of#72270106, #72270155, C3×#404015, …
maximal normal subgroup of#72270106, #72270155, C3×#404015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3756);