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C3×#30434 #10649041

order768
nilpotency class3
isoclinismisoclinic class of #30434
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC3×#30434

Subgroups

subgroupG/H
center, socleC2×C3#2668
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#30434
maximal subgroup ofD3×#30434, C2×C3×#30434
maximal normal subgroup ofD3×#30434, C2×C3×#30434
maximal quotient ofC2×C3×#30434, C3×#421068, C3×#383145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 23422);