Finite Group Databasebeta

← Prev (#10660016) Next (#10660018) →

C3×#41410 #10660017

order768
nilpotency class2
isoclinismisoclinic class of #40484
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4

Aliases

direct productC3×#41410

Subgroups

subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

2630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 5 generators on 23 points:

(1,2,3)(5,7)(8,9)(10,15)(11,14)(12,13)(16,17,20,18)(19,22,23,21);(9,13)(11,15)(17,18)(21,22);(4,5,6,7)(9,13)(10,14)(16,18,20,17)(19,22,23,21);(8,10,12,14)(9,11,13,15)(16,19,20,23)(17,21,18,22);(8,11)(9,14)(10,13)(12,15)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253717, D3×#41410
maximal subgroup of#72253717, D3×#41410, C3×#7918389
maximal normal subgroup of#72253717, D3×#41410, C3×#7918389
maximal quotient ofC3×#4421184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 34398);