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C3×#12523 #10631130

order768
nilpotency class4
isoclinismisoclinic class of #12503
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C8

Aliases

direct productC3×#12523

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3#493
commutator, Frattini, cosocleC4×C8C23×C3
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G

262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,5,8,11)(6,12,15,19)(7,16,13,10)(9,17,18,14)(21,25)(22,29)(23,30)(26,31)(27,35)(28,34)(32,33);(4,6,13,17,8,15,7,14)(5,9,16,19,11,18,10,12);(20,21,26,35,23,27,22,25,24,28,29,33,30,32,31,34)

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#491500, D3×#12523, #72225953, …
maximal normal subgroup ofC3×#491500, D3×#12523, #72225953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5511);