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C22×C3×#575 #10681690

order768
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C26×C3 · Schur multiplier (H2) = C214

Aliases

direct productC22×C3×#575

Subgroups

subgroupG/H
centerC22×C3×C4C2×D4
commutator, Frattini, cosocleC4C26×C3
socleC23×C3C22×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C24×C3×C4 ◁ G

11294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 6 generators on 23 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC22×#87200, C22×D3×#575
maximal subgroup ofC22×#87200, C22×D3×#575, C22×C3×#3585
maximal normal subgroup ofC22×#87200, C22×D3×#575, C22×C3×#3585

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 56071);