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C3×#394602 #72270366

order1536
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#394602

Subgroups

subgroupG/H
center, socleC22×C3C2×#394
commutator, Frattini, cosocleC23×C4C24×C3
prime radicalC2×C3×#3003C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3×C4 ◁ G

5398 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)(8,9,11,16,13,17,12,15)(10,18,20,14,19,22,21,23);(4,5)(6,7)(8,10,12,21,13,19,11,20)(9,14,16,22,17,23,15,18);(5,7);(5,7)(9,15)(10,19)(11,12)(14,22)(16,17)(18,23)

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#394602, #72271329
maximal subgroup ofD3×#394602, #72271329, C3×#72215520
maximal normal subgroup ofD3×#394602, #72271329, C3×#72215520

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3)(5,7)(8,9,11,16,13,17,12,15)(10,18,20,14,19,22,21,23), (4,5)(6,7)(8,10,12,21,13,19,11,20)(9,14,16,22,17,23,15,18), (5,7), (5,7)(9,15)(10,19)(11,12)(14,22)(16,17)(18,23));