Finite Group Databasebeta

← Prev (#66114) Next (#66116) →

Cayley diagram

C22×#681 #66115

order320
isoclinismisoclinic class of Dic10
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC22×#681

Splittings

SylowC5 ⋊ C22×M16

Subgroups

subgroupG/H
centerC22×C4C2×D5
commutatorC2×C5C23×C4
FrattiniC4C23×D5
FittingC23×C4×C5C2
socleC23×C5C2×C4
prime radicalC24×C5C4
cosocleC4×C5C24
hypercenterC23×C4D5
hypocenterC5C22×M16
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ C23×C4

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10600241, #10600243, #10600246, …
maximal normal subgroup of#10600241, #10600243, #10600246, …
maximal quotient ofC22×#3935, C2×C4×#681, C22×C4⋊#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1453);