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C3×#10755 #10629362

order768
nilpotency class3
isoclinismisoclinic class of #1913
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C26

Aliases

direct productC3×#10755

Subgroups

subgroupG/H
center, socleC22×C3#386
commutatorC23C2×C3×C42
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×#41C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

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

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253906, #72253915, D3×#10755
maximal subgroup of#72253906, #72253915, C3×#147813, …
maximal normal subgroup of#72253906, #72253915, C3×#147813, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3743);