Finite Group Databasebeta

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C3×#62818 #10681425

order768
nilpotency class2
isoclinismisoclinic class of #62817
automorphism?
homologyAbelianization (H1) = C26×C3 · Schur multiplier (H2) = C213

Aliases

direct productC3×#62818

Subgroups

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

8042 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)(5,8)(9,11)(12,13)(14,15)(16,17)(18,19);(6,10)(9,11)(13,17)(14,18);(4,5)(6,9)(7,8)(10,11)(12,14,16,18)(13,15,17,19);(12,15)(13,18)(14,17)(16,19);(4,6)(5,9)(7,10)(8,11);(4,5)(6,9)(7,8)(10,11)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#62818, #72271112, #72271210, …
maximal subgroup ofD3×#62818, #72271112, C3×#6512843, …
maximal normal subgroup ofD3×#62818, #72271112, C3×#6512843, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 55806);