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C3×#417674 #72253284

order1536
nilpotency class3
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#417674

Subgroups

subgroupG/H
center, socleC22×C3#2668
commutator, Frattini, cosocleC23×C4C24×C3
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

5826 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)(5,6,8,14,10,12,9,13)(7,15,18,11,16,19,17,20)(21,22,23);(5,7,8,17,10,16,9,18)(6,11,14,15,12,20,13,19);(2,4)(6,12)(13,14);(6,13)(7,16)(8,9)(11,15)(12,14)(19,20)

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#417674, #72255039, #72255104
maximal subgroup ofD3×#417674, #72255039, #72255104, …
maximal normal subgroup ofD3×#417674, #72255039, #72255104, …

GAP

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

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