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C3×#511811 #72272068

order1536
nilpotency class4
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C28

Aliases

direct productC3×#511811

Subgroups

subgroupG/H
center, socleC2×C3#32898
commutator, Frattini, cosocleC22×D4C24×C3
LCSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C23×C3×D4 ◁ G

5518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

(1,2,3)(4,5)(6,9)(7,10)(8,11)(12,18)(13,17)(14,16)(15,19);(4,6,8,14)(5,9,11,16)(7,12,15,13)(10,17,19,18);(9,16)(17,18);(6,12,14,13)(9,17,16,18)

Conjugacy classes & character table

159 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72272080, D3×#511811
maximal subgroup of#72272080, D3×#511811, C3×#72216366
maximal normal subgroup of#72272080, D3×#511811, C3×#72216366

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