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C3×#20984 #10639591

order768
nilpotency class3
isoclinismisoclinic class of #17386
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productC3×#20984

Subgroups

subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#534C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×D4 ◁ G

2102 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)(4,5)(6,7)(8,9)(10,13,17,20)(11,14)(12,15)(16,19,22,23)(18,21);(8,10)(9,13,15,20)(11,16)(12,17)(14,19,21,23)(18,22);(5,7)(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,22)(20,23);(11,18)(14,21)(16,22)(19,23)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270125, #72270135, D3×#20984
maximal subgroup of#72270125, #72270135, C3×#404015, …
maximal normal subgroup of#72270125, #72270135, C3×#404015, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13972);