Finite Group Databasebeta

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C3×#15674 #10634281

order768
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4

Aliases

direct productC3×#15674

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#15674, #72254561, #72255197
maximal subgroup ofD3×#15674, #72254561, C3×#619101, …
maximal normal subgroup ofD3×#15674, #72254561, C3×#619101, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8662);