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C3×#4326 #10587175

order528
isoclinismisoclinic class of #4323
automorphismAut(G) = C22×D4×AGL(1,𝔽11) (order 3520) · Inn(G) = C22⋊D11 · Out(G) = C23×C5
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×#4326

Splittings

HallC3×C11 ⋊ Q16
SylowC11 ⋊ C3×Q16

Subgroups

subgroupG/H
centerC2×C3C22⋊D11
commutator, cosocleC4×C11C22×C3
FrattiniC4C2×C3×D11
FittingC3×Q8×C11C2
socle, prime radicalC2×C3×C11D4
hypercenterC3×Q8D11
hypocenterC11C3×Q16
LCSG ▷ C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 528

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,3)(4,5,7,9,11,13,14,12,10,8,6)(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23);(5,6)(7,8)(9,10)(11,12)(13,14)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#4326, #11760960, #11760961, …
maximal normal subgroup ofD3×#4326, #11760960, #11760961, …
maximal quotient ofC3×#66871, C3×#66874, C3×#66899, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 45);