Finite Group Databasebeta

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Cayley diagram

C2×C3×#713 #92067

order480
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productC2×C3×#713

Splittings

FittingC2×C3×C5×Q8 ⋊ C2
HallC3×C5 ⋊ C2×#41
SylowC5 ⋊ C2×C3×#41

Subgroups

subgroupG/H
centerC22×C3C22×D5
commutator, cosocleC2×C5C24×C3
FrattiniC2C23×C3×D5
FittingC2×C3×C5×Q8C2
socleC22×C3×C5C23
hypercenterC2×C3×Q8D5
hypocenterC5C2×C3×#41
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

328 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 240

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11749198, #11749207, #11749270, …
maximal normal subgroup of#11749198, #11749207, #11749270, …
maximal quotient ofC2×C3×#4036, C2×C3×#4038, C2×C3×#4039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 1143);