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C3×#20925 #10639532

order768
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC3×#20925

Subgroups

subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

2234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271769, #72271772
maximal subgroup ofC3×#359576, #72271769, #72271772
maximal normal subgroup ofC3×#359576, #72271769, #72271772

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13913);