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C3×#20953 #10639560

order768
nilpotency class3
isoclinismisoclinic class of #3019
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC3×#20953

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#535C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1974 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)(5,7)(8,9,12,17,13,15,11,16)(10,18,21,14,20,23,19,22);(4,5,6,7)(8,10)(9,14)(11,21)(12,19)(13,20)(15,22)(16,23)(17,18);(9,15)(11,12)(14,22)(19,21);(9,15)(14,22)(16,17)(18,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253693, #72253705, #72270195, …
maximal subgroup ofC3×#6437445, #72253693, #72253705, …
maximal normal subgroup ofC3×#6437445, #72253693, #72253705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13941);