Finite Group Databasebeta

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Cayley diagram

D5×C22⋊C4 #4026

order160
automorphismAut(G) = AGL(1,𝔽5)×#2846 (order 2560) · Inn(G) = C22×D5 · Out(G) = C2×#120
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

direct productD5×C22⋊C4

Splittings

FittingC5×C22⋊C4 ⋊ C2
SylowC5 ⋊ C2×C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×D5
commutatorC2×C5C22×C4
FittingC5×C22⋊C4C2
socle, cosocleC22×C5C23
prime radicalC23×D5C2
hypercenterC22⋊C4D5
hypocenterC5C2×C22⋊C4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22⋊C4

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

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

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64866, D5×#99, C22⋊C4×AGL(1,𝔽5), …
maximal normal subgroup of#64866, D5×#99, C22⋊C4×AGL(1,𝔽5), …
maximal quotient of#64935, D5×#95, #64953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 101);