Finite Group Databasebeta

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Cayley diagram

D4×#168 #65299

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

Aliases

direct productD4×#168

Splittings

SylowC5 ⋊ C8×D4

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×D5
commutatorC2×C5C22×C8
FittingC4×C5×D4C2
socleC22×C5C22×C4
prime radicalC2×C5×D4C4
cosocleC2×C4×C5C23
hypercenterC4×D4D5
hypocenterC5C8×D4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×D4

134 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)(10,11,12,14,13,15,16,17);(1,2,4,5,3)(7,9);(6,7)(8,9)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4⋊#674, #10598941, #10598942, …
maximal normal subgroup ofD4⋊#674, #10598941, #10598942, …
maximal quotient ofC22⋊C4×#168, #10600242, #10600244, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 637);