Finite Group Databasebeta

← Prev (#10631188) Next (#10631190) →

C3×#12582 #10631189

order768
nilpotency class4
isoclinismisoclinic class of #2127
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC3×#12582

Subgroups

subgroupG/H
center, socleC22×C3C2×#99
commutatorC2×C4C2×C3×C42
Frattini, cosocleC22×D4C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×D4 ◁ G

1026 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)(8,9)(10,16,17,22)(11,19,20,14)(12,15)(13,18)(21,23);(4,5,6,7)(8,10)(9,13,15,21)(11,18)(12,17)(14,16,19,22)(20,23);(10,17)(13,21)(16,22)(18,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#12582, #72271334
maximal subgroup ofC3×#146479, D3×#12582, #72271334
maximal normal subgroup ofC3×#146479, D3×#12582, #72271334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5570);