Finite Group Databasebeta

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Cayley diagram

C3×#401 #5275

order192
nilpotency class2
isoclinismisoclinic class of #392
automorphismAut(G) order 21504
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×#401

Subgroups

subgroupG/H
center, socle, prime radicalC23×C3C23
commutator, Frattini, cosocleC23C23×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

118 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)(3,5,7,8)(9,10,12,14)(11,13,15,16)(17,18,20,21)(19,23,24,22)(25,26,27);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(18,21)(19,24);(2,6)(5,8)(11,15)(13,16)(17,19,20,24)(18,22,21,23)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#78409, #78410, #78411, …
maximal subgroup of#78409, #78410, #78411, …
maximal normal subgroup of#78409, #78410, #78411, …
maximal quotient ofC3×#1443, C3×#2104, C2×C3×#401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 834);