Finite Group Databasebeta

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C3×#14473 #10633080

order768
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26×C4

Aliases

direct productC3×#14473

Subgroups

subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2134 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)(20,21,24,22)(23,26,27,25);(13,16)(17,19)(21,22)(23,27);(4,5)(6,7)(8,9)(10,11)(12,13)(14,17)(15,16)(18,19)(20,22,24,21)(23,26,27,25);(4,5)(6,7)(8,9)(10,11)(12,14,15,18)(13,17,16,19)(20,23,24,27)(21,25,22,26)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253503, D3×#14473, #72253653
maximal subgroup ofC3×#581603, #72253503, D3×#14473, …
maximal normal subgroup ofC3×#581603, #72253503, D3×#14473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7461);