Finite Group Databasebeta

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C3×#418876 #72253359

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

Aliases

direct productC3×#418876

Subgroups

subgroupG/H
center, socleC22×C3#2668
commutator, Frattini, cosocleC23×C4C24×C3
prime radicalC2×C3×#3003C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

4956 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)(5,6,8,12,9,11,15,13)(7,14,18,20,16,19,17,10)(21,22,23);(2,4)(5,7,15,17,9,16,8,18)(6,10,13,19,11,20,12,14);(2,4)(6,11)(12,13);(6,12)(7,16)(8,15)(10,19)(11,13)(14,20)

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253363, D3×#418876
maximal subgroup of#72253363, D3×#418876, C3×#72215520
maximal normal subgroup of#72253363, D3×#418876, C3×#72215520

GAP

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

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