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Cayley diagram

A4×C13⋊C3 #90857

order468
isoclinismisoclinic class of A4×C13⋊C3
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productA4×C13⋊C3
unambiguous semidirect product(C22×C13⋊C3)⋊C3
unambiguous semidirect product(A4×C13)⋊C3
unambiguous semidirect product(C22⋊(C13⋊C3))⋊C3

Splittings

Fitting, socle, derived, HallC22×C13 ⋊ C32
SylowC13 ⋊ C3×A4
SylowC22 ⋊ C3×C13⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C13C32
LCSG ▷ C22×C13
UCSC1

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

(1,4)(2,3)(6,7,9)(8,11,14)(10,13,12)(15,17,16);(2,3,4)(5,6,8,7,10,14,11,15,17,9,12,16,13)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×C13⋊C3, A4⋊(D13⋊C3), A4×D13⋊C3, …
maximal normal subgroup ofS4×C13⋊C3, A4⋊(D13⋊C3), A4×D13⋊C3, …
maximal quotient ofSL(2,𝔽3)×C13⋊C3, C2×A4×C13⋊C3, C22⋊C9×C13⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(468, 32);