Finite Group Databasebeta

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Cayley diagram

D4×#170 #65772

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

Aliases

direct productD4×#170

Splittings

SylowC5 ⋊ C8×D4

Subgroups

subgroupG/H
center, FrattiniC22C22×AGL(1,𝔽5)
commutatorC2×C5C22×C8
Fitting, prime radicalC2×C5×D4C4
socleC22×C5C22×C4
cosocleC2×Dic5C23
hypercenterC2×D4AGL(1,𝔽5)
hypocenterC5C8×D4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×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);(1,2,3,5,4)(7,9);(6,7)(8,9)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10603935, #10603936, #10603937, …
maximal normal subgroup of#10603935, #10603936, #10603937, …
maximal quotient ofC22⋊C4×#170, #10604231, #10604237, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1110);