Finite Group Databasebeta

← Prev (#10642516) Next (#10642518) →

C3×#23910 #10642517

order768
nilpotency class3
isoclinismisoclinic class of #23906
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#23910

Subgroups

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

1866 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)(10,16)(12,20)(14,22)(15,19)(17,23)(18,21);(4,5)(6,7)(8,9,13,11)(10,14,18,17)(12,19,20,15)(16,23,21,22);(5,7)(8,10)(9,14)(11,17)(12,21)(13,18)(15,23)(16,20)(19,22);(8,11,13,9)(10,17,18,14)(12,15,20,19)(16,22,21,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271495, #72271499, #72271501
maximal subgroup of#72271495, #72271499, #72271501, …
maximal normal subgroup of#72271495, #72271499, #72271501, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16898);