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Cayley diagram

C2×A5⋊C4 #91876

order480
isoclinismisoclinic class of S5
automorphismAut(G) = D4×S5 (order 960) · Inn(G) = S5 · Out(G) = D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22

Aliases

direct productC2×A5⋊C4
unambiguous semidirect product(C2×A5)⋊C4

Splittings

derivedA5 ⋊ C2×C4

Subgroups

subgroupG/H
center, Fitting, hypercenterC22S5
commutator, hypocenterA5C2×C4
FrattiniC2C2×S5
socle, prime radicalC22×A5C2
cosocleC2×A5C22
LCSG ▷ A5
UCSC1 ◁ C22

117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

(3,6,4,5)(8,9,10,11);(1,2)(3,4)(5,6)(7,10,9)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4⋊(A5⋊C4), C4×A5⋊C4, (C2×S5)⋊C4, …
maximal normal subgroup ofC4⋊(A5⋊C4), C4×A5⋊C4, (C2×S5)⋊C4, …
maximal quotient of#11751496, C2×A5⋊C8, C2×GL(2,𝔽5), …
outer automorphism group of#1039, C2×C5⋊D5, C3×C5⋊D5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 952);