Finite Group Databasebeta

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C3×#35750 #10654357

order768
nilpotency class2
isoclinismisoclinic class of #35680
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C210

Aliases

direct productC3×#35750

Subgroups

subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

5482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#35750, #72270689
maximal subgroup ofD3×#35750, #72270689, C3×#7695992
maximal normal subgroup ofD3×#35750, #72270689, C3×#7695992
maximal quotient ofC3×#4306523, C3×#517592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 28738);