Finite Group Databasebeta

← Prev (#10643004) Next (#10643006) →

C3×#24398 #10643005

order768
nilpotency class3
isoclinismisoclinic class of #24398
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#24398

Subgroups

subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1476 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)(4,5)(6,7)(8,9,12,16,11,15,21,14)(10,17,18,23,19,13,20,22);(5,7)(8,10)(9,13)(11,19)(12,18)(14,23)(15,17)(16,22)(20,21);(9,14)(10,18)(12,21)(15,16)(19,20)(22,23);(5,7)(9,15)(13,17)(14,16)(22,23)

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#24398, #72270487
maximal subgroup ofD3×#24398, #72270487, C3×#6437793
maximal normal subgroup ofD3×#24398, #72270487, C3×#6437793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17386);