Finite Group Databasebeta

← Prev (#91179) Next (#91181) →

Cayley diagram

C5×#859 #91180

order480
isoclinismisoclinic class of #226
automorphismAut(G) = C22×C4×S4 (order 384) · Inn(G) = S4 · Out(G) = C22×C4
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×#859

Splittings

derivedSL(2,𝔽3) ⋊ C4×C5

Subgroups

subgroupG/H
center, socle, hypercenterC22×C5S4
commutator, hypocenterSL(2,𝔽3)C4×C5
FrattiniC22C5×S4
FittingC2×C5×Q8D3
prime radicalC2×C5×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C2×C5
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C5

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751587, #11751589, #11751591, …
maximal normal subgroup of#11751587, #11751589, #11751591, …
maximal quotient ofC5×SL(2,𝔽3)⋊C8, C2×C5×#859, C5×#63421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 256);