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C3×#24399 #10643006

order768
nilpotency class3
isoclinismisoclinic class of #24398
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productC3×#24399

Subgroups

subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 96

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

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

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72270461, D3×#24399
maximal subgroup of#72270461, D3×#24399, C3×#6437793
maximal normal subgroup of#72270461, D3×#24399, C3×#6437793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 17387);