Finite Group Databasebeta

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Cayley diagram

C4×C22⋊D5 #4074

order160
isoclinismisoclinic class of Dic10
automorphismAut(G) = C26×AGL(1,𝔽5) (order 1280) · Inn(G) = C2×D5 · Out(G) = C24×C4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC4×C22⋊D5

Splittings

FittingC22×C4×C5 ⋊ C2
SylowC5 ⋊ C4×D4

Subgroups

subgroupG/H
centerC2×C4C2×D5
commutatorC2×C5C22×C4
FrattiniC22C22×D5
FittingC22×C4×C5C2
socle, cosocleC22×C5C23
prime radicalC2×C22⋊D5C2
hypercenterC22×C4D5
hypocenterC5C4×D4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C22×C4

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 3 generators on 13 points:

(1,2,4,5,3)(11,13);(2,3)(4,5)(10,11)(12,13);(6,7,8,9)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#65014, #65017, #65018, …
maximal normal subgroup of#65014, #65017, #65018, …
maximal quotient ofC4×#683, #65222, C4×#685, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 149);