Finite Group Databasebeta

← Prev (#10642518) Next (#10642520) →

C3×#23912 #10642519

order768
nilpotency class3
isoclinismisoclinic class of #23912
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#23912

Subgroups

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

1950 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)(8,9)(10,17)(11,15)(12,20)(13,16)(14,18)(19,22)(21,23);(4,5)(6,7)(11,12)(15,20)(17,22)(18,21);(5,7)(8,10)(9,14)(11,18)(12,21)(13,19)(15,22)(16,23)(17,20);(8,11)(9,15)(10,18)(12,13)(14,22)(16,20)(17,23)(19,21)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253700, D3×#23912, #72270304
maximal subgroup ofC3×#6437445, #72253700, C3×#501097, …
maximal normal subgroup ofC3×#6437445, #72253700, C3×#501097, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16900);