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D264 #10587196

order528
automorphismAut(G) = D3×AGL(1,ℤ/8)×AGL(1,𝔽11) (order 21120) · Inn(G) = D132 · Out(G) = C24×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD264

Splittings

FittingC3×C8×C11 ⋊ C2
HallC3×C11 ⋊ D8
SylowC11 ⋊ D24
SylowC3 ⋊ D88

Subgroups

subgroupG/H
centerC2D132
commutator, cosocleC3×C4×C11C22
FrattiniC4C2×D33
FittingC3×C8×C11C2
socleC2×C3×C11D4
hypercenterC8D33
hypocenterC3×C11D8
LCSG ▷ C3×C4×C11 ▷ C2×C3×C11 ▷ C3×C11
UCSC1 ◁ C2 ◁ C4 ◁ C8

44 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)(12,13,14,15,16,17,18,19,20,21,22);(2,8)(3,7)(4,6)(10,11)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

2^4,3,11|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^10

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11760475, #11760476, #11760480, …
maximal normal subgroup of#11760475, #11760476, #11760480, …
maximal quotient of#11760620, D528, Dic264, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(528, 66);