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Cayley diagram

C4×C8×D5 #64973

order320
isoclinismisoclinic class of D5
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4

Aliases

direct productC4×C8×D5

Splittings

FittingC4×C5×C8 ⋊ C2
derived, SylowC5 ⋊ C2×C4×C8

Subgroups

subgroupG/H
center, hypercenterC4×C8D5
commutator, hypocenterC5C2×C4×C8
FrattiniC2×C4C22×D5
FittingC4×C5×C8C2
socleC22×C5C22×C4
prime radicalC22×D5C2×C4
cosocleC2×C4×C5C23
LCSG ▷ C5
UCSC1 ◁ C4×C8

162 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:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10598530, #10598531, C82×D5, …
maximal normal subgroup of#10598530, #10598531, C82×D5, …
maximal quotient ofC82×D5, C8×#676, #10598840, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 311);