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Cayley diagram

C3×#499 #5373

order192
nilpotency class3
isoclinismisoclinic class of #465
automorphismAut(G) = C2×#72215520 (order 2048) · Inn(G) = C2×D4 · Out(G) = C22×#120
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×#499

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C2×D4
commutator, Frattini, cosocleC2×C4C23×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ G

86 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19)(25,26,27);(1,3,4,8)(2,5,6,7)(10,14)(15,22)(16,21)(17,24);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#67799, #67800, #71351, …
maximal subgroup of#67799, #67800, C3×#1363, …
maximal normal subgroup of#67799, #67800, C3×#1363, …
maximal quotient ofC3×#1846, C3×#1941, C3×#1942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 932);