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C3×#23445 #10642052

order768
nilpotency class3
isoclinismisoclinic class of #23444
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C4

Aliases

direct productC3×#23445

Subgroups

subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

2282 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)(11,14)(12,15)(16,20)(17,19)(18,21)(22,23);(11,18)(13,19)(14,21)(16,22);(5,7)(8,10)(9,13)(11,16)(12,17)(14,20)(15,19)(18,22)(21,23);(8,11,12,18)(9,14,15,21)(10,16,17,22)(13,20,19,23)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270651, D3×#23445
maximal subgroup ofC3×#6431795, #72270651, D3×#23445
maximal normal subgroup ofC3×#6431795, #72270651, D3×#23445

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 16433);