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C3×#14169 #10632776

order768
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C43

Aliases

direct productC3×#14169

Subgroups

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

2014 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)(8,9,11,10)(12,13,16,14)(15,18,19,17);(13,14)(17,18)(20,21,23,25)(22,24,26,27);(4,5,6,7)(8,10,11,9)(12,14,16,13)(15,18,19,17)(20,21)(22,24)(23,25)(26,27);(12,15,16,19)(13,17,14,18)(20,22,23,26)(21,24,25,27)

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72253481, D3×#14169, #72253636, …
maximal subgroup ofC3×#581603, #72253481, D3×#14169, …
maximal normal subgroup ofC3×#581603, #72253481, D3×#14169, …
maximal quotient ofC3×#375718

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7157);