Finite Group Databasebeta

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C3×#61066 #10679673

order768
nilpotency class3
isoclinismisoclinic class of D4×AGL(1,ℤ/8)
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC3×#61066

Subgroups

subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×D4 ◁ G

2912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 23 points:

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

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofD3×#61066, #72253892
maximal subgroup ofD3×#61066, #72253892, C3×#10576108
maximal normal subgroup ofD3×#61066, #72253892, C3×#10576108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54054);