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C3×#20945 #10639552

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

Aliases

direct productC3×#20945

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1862 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)(8,9,13,16,14,10,12,17)(11,19,21,18,22,23,20,15);(4,5,6,7)(8,10)(9,14)(11,18)(12,17)(13,16)(15,22)(19,21)(20,23);(8,11)(9,15)(10,18)(12,20)(13,21)(14,22)(16,19)(17,23);(8,12,14,13)(9,16,10,17)(11,20,22,21)(15,19,18,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248607, #72248612, D3×#20945, …
maximal subgroup of#72248607, #72248612, C3×#364592, …
maximal normal subgroup of#72248607, #72248612, C3×#364592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13933);