Finite Group Databasebeta

← Prev (#10661701) Next (#10661703) →

C3×#43095 #10661702

order768
nilpotency class2
isoclinismisoclinic class of #42996
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#43095

Subgroups

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

3420 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)(10,14)(13,15)(16,17,20,21)(18,19,23,22);(8,9)(10,15)(11,12)(13,14)(16,17)(18,22)(19,23)(20,21);(4,5,6,7)(9,12)(13,15)(17,21)(18,23);(8,9)(10,13)(11,12)(14,15)(16,18)(17,19)(20,23)(21,22);(8,10)(9,13)(11,14)(12,15)(16,19)(17,18)(20,22)(21,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248080, D3×#43095
maximal subgroup of#72248080, D3×#43095, C3×#7849701
maximal normal subgroup of#72248080, D3×#43095, C3×#7849701
maximal quotient ofC3×#1418391, C3×#1260433

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 36083);