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C3×#16362 #10634969

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

Aliases

direct productC3×#16362

Subgroups

subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC22×C3×#121C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

2226 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)(20,21,24,23)(22,26,27,25);(14,18)(16,19)(20,22,24,27)(21,25,23,26);(8,9)(10,11)(12,14)(13,16)(15,18)(17,19)(20,22)(21,26)(23,25)(24,27);(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)(20,23,24,21)(22,26,27,25)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270779, D3×#16362, #72270917, …
maximal subgroup of#72270779, D3×#16362, C3×#1260433, …
maximal normal subgroup of#72270779, D3×#16362, C3×#1260433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 9350);