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Cayley diagram

C11×#168 #89150

order440
isoclinismisoclinic class of D5
automorphismAut(G) = C23×C5×AGL(1,𝔽5) (order 800) · Inn(G) = D5 · Out(G) = C24×C5
homologyAbelianization (H1) = C8×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×#168

Splittings

derived, SylowC5 ⋊ C8×C11
HallC5×C11 ⋊ C8

Subgroups

subgroupG/H
center, hypercenterC4×C11D5
commutator, hypocenterC5C8×C11
FrattiniC4D5×C11
FittingC4×C5×C11C2
socle, prime radicalC2×C5×C11C4
cosocleC4×C5C2×C11
LCSG ▷ C5
UCSC1 ◁ C4×C11

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 440

A minimal faithful permutation representation — 2 generators on 24 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC11×#674, C11⋊#674, D11×#168, …
maximal normal subgroup ofC11×#674, C11⋊#674, D11×#168, …
maximal quotient ofC11×#672, C2×C11×#168, C3×C11×#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(440, 3);