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C3×#14662 #10633269

order768
nilpotency class2
isoclinismisoclinic class of #2812
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C43

Aliases

direct productC3×#14662

Subgroups

subgroupG/H
center, socleC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

1854 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)(12,13,15,17)(14,16,18,19)(20,21,24,23)(22,25,26,27);(5,7)(8,9,10,11)(14,18)(16,19)(21,23)(22,26);(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)(20,22)(21,25)(23,27)(24,26);(12,14)(13,16)(15,18)(17,19)(20,23,24,21)(22,25,26,27)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253493, D3×#14662, #72253641, …
maximal subgroup ofC3×#581603, #72253493, D3×#14662, …
maximal normal subgroup ofC3×#581603, #72253493, D3×#14662, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7650);