Finite Group Databasebeta

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C3×#14825 #10633432

order768
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#14825

Subgroups

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

2198 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)(12,13,15,17)(14,16,18,19)(21,24)(22,26);(14,18)(16,19)(20,21,23,24)(22,25,26,27);(8,9)(10,11)(12,14)(13,16)(15,18)(17,19)(20,21)(22,25)(23,24)(26,27);(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)(20,22,23,26)(21,25,24,27)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270785, D3×#14825, #72270845, …
maximal subgroup of#72270785, D3×#14825, C3×#1260433, …
maximal normal subgroup of#72270785, D3×#14825, C3×#1260433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7813);