Finite Group Databasebeta

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Cayley diagram

C22×#168 #4066

order160
isoclinismisoclinic class of D5
automorphismAut(G) = C2×AGL(1,𝔽5)×#5934 (order 7680) · Inn(G) = D5 · Out(G) = C22×#5934
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC22×#168

Splittings

derived, SylowC5 ⋊ C22×C8

Subgroups

subgroupG/H
center, hypercenterC22×C4D5
commutator, hypocenterC5C22×C8
FrattiniC4C22×D5
FittingC22×C4×C5C2
socle, prime radicalC23×C5C4
cosocleC4×C5C23
LCSG ▷ C5
UCSC1 ◁ C22×C4

76 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)(6,7)(8,9)(10,11,12,14,13,15,16,17);(3,4)(5,6,8,9,7);(1,2)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64744, #64749, #64770, …
maximal normal subgroup of#64744, #64749, #64770, …
maximal quotient ofC2×C4×#168, C2×C4⋊#168, #65214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 141);