Finite Group Databasebeta

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C3×#16632 #10635239

order768
nilpotency class2
isoclinismisoclinic class of #16249
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4

Aliases

direct productC3×#16632

Subgroups

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

1966 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)(12,13,15,17)(14,18,19,16)(21,25)(22,26);(12,13)(14,16)(15,17)(18,19)(20,21,24,25)(22,27,26,23);(8,9)(10,11)(12,13)(14,18)(15,17)(16,19)(20,22,24,26)(21,23,25,27);(4,5)(6,7)(8,9)(10,11)(12,14)(13,16)(15,19)(17,18)(20,23)(21,22)(24,27)(25,26)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72255513, #72255518, #72270099, …
maximal subgroup of#72255513, #72255518, C3×#1262353, …
maximal normal subgroup of#72255513, #72255518, C3×#1262353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 9620);