Finite Group Databasebeta

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Cayley diagram

C22×C3⋊#168 #91809

order480
isoclinismisoclinic class of D15
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C23

Aliases

direct productC22×C3⋊#168

Splittings

derived, HallC3×C5 ⋊ C22×C8
SylowC5 ⋊ C22×C3⋊C8
SylowC3 ⋊ C22×#168

Subgroups

subgroupG/H
center, hypercenterC22×C4D15
commutator, hypocenterC3×C5C22×C8
FrattiniC4C22×D15
FittingC22×C3×C4×C5C2
socle, prime radicalC23×C3×C5C4
cosocleC3×C4×C5C23
LCSG ▷ C3×C5
UCSC1 ◁ C22×C4

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 480

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(6,7)(9,10)(11,12)(13,14,15,17,16,18,19,20);(3,4)(5,6,7)(8,9,11,12,10);(1,2)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11745966, #11745967, #11745979, …
maximal normal subgroup of#11745966, #11745967, #11745979, …
maximal quotient ofC2×C4×C3⋊#168, C2×C4⋊(C3⋊#168), #11751026, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 885);