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Cayley diagram

D5×C4⋊C8 #65121

order320
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productD5×C4⋊C8

Splittings

FittingC5×C4⋊C8 ⋊ C2
SylowC5 ⋊ C2×C4⋊C8

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×D5
commutatorC2×C5C22×C8
FittingC5×C4⋊C8C2
socleC22×C5C22×C4
prime radicalC22×D5C2×C4
cosocleC2×C4×C5C23
hypercenterC4⋊C8D5
hypocenterC5C2×C4⋊C8
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4⋊C8

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 3 generators on 17 points:

(1,2,3,4)(13,14,16,17,15);(2,4)(5,6,7,9,8,10,11,12);(14,15)(16,17)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598586, #10598587, #10598589, …
maximal normal subgroup of#10598586, #10598587, #10598589, …
maximal quotient ofD5×#334, #10599320, #10599321, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 459);