Finite Group Databasebeta

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C3×(C11⋊C5)⋊Q8 #13050319

order1320
isoclinismisoclinic class of (C11⋊C5)⋊Q8
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC3×(C11⋊C5)⋊Q8
unambiguous semidirect product(C11⋊C5)⋊(C3×Q8)
unambiguous semidirect productDic22⋊(C3×C5)
unambiguous semidirect product(C3×Dic22)⋊C5

Splittings

HallC3×Dic22 ⋊ C5
HallC3×C11⋊C5 ⋊ Q8
HallDic22 ⋊ C3×C5
HallC11⋊C5 ⋊ C3×Q8
HallC3×C11 ⋊ C5×Q8
SylowC11 ⋊ C3×C5×Q8

Subgroups

subgroupG/H
centerC2×C3C2×AGL(1,𝔽11)
commutator, cosocleC2×C11C22×C3×C5
FrattiniC2C2×C3×AGL(1,𝔽11)
FittingC3×C4×C11C2×C5
socleC2×C3×C11C22×C5
prime radicalC2×C3×C11⋊C5C22
hypercenterC3×C4AGL(1,𝔽11)
hypocenterC11C3×C5×Q8
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C3 ◁ C3×C4

48 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)(4,6,7,11)(5,8,9,10)(12,13,14,19,22)(15,20,16,21,18);(4,5,7,9)(6,10,11,8)(13,14)(15,16)(17,18)(19,21)(20,22)

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#4314⋊C5, C3×#4324⋊C5, #72216820, …
maximal normal subgroup ofC3×#4314⋊C5, C3×#4324⋊C5, #72216820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 23);