Finite Group Databasebeta

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C3×#16600 #10635207

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

Aliases

direct productC3×#16600

Subgroups

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

2010 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)(9,11)(13,16)(14,18)(21,25)(23,26);(12,13)(14,17)(15,16)(18,19)(20,21,24,25)(22,26,27,23);(8,9)(10,11)(12,13)(14,19)(15,16)(17,18)(20,22,24,27)(21,23,25,26);(4,5)(6,7)(8,9)(10,11)(12,14)(13,17)(15,18)(16,19)(20,23)(21,22)(24,26)(25,27)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72255511, #72255522, #72270123, …
maximal subgroup of#72255511, #72255522, C3×#1262353, …
maximal normal subgroup of#72255511, #72255522, C3×#1262353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 9588);