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C11×(C2×D3)⋊C4 #10587187

order528
isoclinismisoclinic class of Dic6
automorphismAut(G) = C25×C5×D3 (order 960) · Inn(G) = C2×D3 · Out(G) = C24×C5
homologyAbelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C22

Aliases

direct productC11×(C2×D3)⋊C4
unambiguous semidirect product(C2×D3)⋊(C4×C11)

Splittings

FittingC2×C3×C4×C11 ⋊ C2
HallC3×C11 ⋊ C22⋊C4
SylowC3 ⋊ C11×C22⋊C4

Subgroups

subgroupG/H
centerC22×C11C2×D3
commutatorC2×C3C2×C4×C11
FrattiniC22C2×D3×C11
FittingC2×C3×C4×C11C2
socleC22×C3×C11C22
prime radicalC22×D3×C11C2
cosocleC22×C3C22×C11
hypercenterC2×C4×C11D3
hypocenterC3C11×C22⋊C4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C11 ◁ C2×C4×C11

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 264

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11760742, #11760744, #11760762, …
maximal normal subgroup of#11760742, #11760744, #11760762, …
maximal quotient ofC11×#805, C11×#806, C11×#809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 57);