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Cayley diagram

C5×GU(2,𝔽3) #91181

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×GU(2,𝔽3)

Splittings

derivedSL(2,𝔽3) ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5S4
commutator, hypocenterSL(2,𝔽3)C4×C5
FrattiniC4C5×S4
FittingC5×#41D3
socleC2×C5SL(2,ℤ/4)
prime radicalC5×#41⋊C3C2
cosocle#41⋊C3C2×C5
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C4×C5

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5)(6,15,11,8)(9,28,18,16)(12,25,14,27)(17,20,26,19)(22,24,29,23);(6,9,17)(7,12,16)(8,14,22)(10,20,25)(11,19,28)(13,24,26)(15,23,27)(18,29,21)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11751616, #11751618, D5×GU(2,𝔽3), …
maximal normal subgroup of#11751616, #11751618, D5×GU(2,𝔽3), …
maximal quotient ofC5×SL(2,𝔽3)⋊C8, C2×C5×GU(2,𝔽3), C5×#63422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 257);