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C3×#59000 #10677607

order768
nilpotency class2
isoclinismisoclinic class of #58990
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC3×#59000

Subgroups

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

3360 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)(9,11)(13,15)(17,18);(5,7)(9,11)(12,13,16,15)(14,17,19,18);(8,9)(10,11)(12,14,16,19)(13,17,15,18);(4,5)(6,7)(8,9)(10,11);(12,15,16,13)(14,18,19,17)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#59000, #72270671
maximal subgroup ofD3×#59000, #72270671, C3×#7695992
maximal normal subgroup ofD3×#59000, #72270671, C3×#7695992
maximal quotient ofC3×#647667

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 51988);