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C3×#15177 #10633784

order768
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C27

Aliases

direct productC3×#15177

Subgroups

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

2770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270773, #72270828, D3×#15177
maximal subgroup of#72270773, #72270828, C3×#647667, …
maximal normal subgroup of#72270773, #72270828, C3×#647667, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 8165);