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C3×#11034 #10629641

order768
nilpotency class3
isoclinismisoclinic class of #2048
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC3×#11034

Subgroups

subgroupG/H
center, socleC22×C3#394
commutatorC22×C4C22×C3×C4
Frattini, cosocleC23×C4C23×C3
prime radicalC22×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#72248371, #72248374, D3×#11034, …
maximal subgroup of#72248371, #72248374, C3×#383145, …
maximal normal subgroup of#72248371, #72248374, C3×#383145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 4022);