Finite Group Databasebeta

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C3×#11058 #10629665

order768
nilpotency class3
isoclinismisoclinic class of #9091
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC3×#11058

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

970 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)(8,9,12,17,13,15,11,16)(10,14,19,21,20,22,18,23);(4,5,6,7)(8,10)(9,14)(11,18)(12,19)(13,20)(15,22)(16,23)(17,21);(9,15)(10,18)(11,12)(14,21)(19,20)(22,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253291, #72253296, #72253351, …
maximal subgroup of#72253291, #72253296, C3×#454694, …
maximal normal subgroup of#72253291, #72253296, C3×#454694, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4046);