Finite Group Databasebeta

← Prev (#10654601) Next (#10654603) →

C3×#35995 #10654602

order768
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#35995

Subgroups

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

2848 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)(4,5,6,7)(16,17,21,19)(18,23,20,22);(4,5)(6,7)(8,9)(10,13)(11,14)(12,15)(16,18,21,20)(17,22,19,23);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,17,21,19)(18,23,20,22);(8,10)(9,13)(11,15)(12,14)(16,19,21,17)(18,23,20,22);(8,9)(10,12)(11,14)(13,15)(16,20,21,18)(17,22,19,23)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#35995, #72254440
maximal subgroup ofC3×#7837029, D3×#35995, #72254440
maximal normal subgroup ofC3×#7837029, D3×#35995, #72254440
maximal quotient ofC3×#4929019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 28983);