Finite Group Databasebeta

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C3×#63012 #10681619

order768
nilpotency class2
isoclinismisoclinic class of #63011
automorphism?
homologyAbelianization (H1) = C26×C3 · Schur multiplier (H2) = C213

Aliases

direct productC3×#63012

Subgroups

subgroupG/H
center, socleC22×C3C26
commutator, Frattini, cosocleC22C26×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

8050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 6 generators on 19 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72271941, #72271944, D3×#63012
maximal subgroup ofC3×#511853, #72271941, #72271944, …
maximal normal subgroup ofC3×#511853, #72271941, #72271944, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 56000);