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C3×C4×#550 #10652767

order768
nilpotency class2
isoclinismisoclinic class of D42
automorphism?
homologyAbelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C29

Aliases

direct productC3×C4×#550

Subgroups

subgroupG/H
centerC22×C3×C4C24
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
socleC23×C3C25
prime radicalC2×C3×#550C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

3420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 384

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

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

Conjugacy classes & character table

300 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C4×#3498
maximal normal subgroup ofC3×C4×#3498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 27148);