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Cayley diagram

AGL(1,𝔽5)×SL(2,𝔽3) #91889

order480
isoclinismisoclinic class of #4133⋊C3
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = A4×AGL(1,𝔽5) · Out(G) = C22
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽5)×SL(2,𝔽3)
unambiguous semidirect product(Q8×AGL(1,𝔽5))⋊C3

Splittings

Fitting, derivedC5×Q8 ⋊ C3×C4
HallQ8×AGL(1,𝔽5) ⋊ C3
SylowC5 ⋊ C4×SL(2,𝔽3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×AGL(1,𝔽5)
commutator, Fitting, hypocenterC5×Q8C3×C4
socleC2×C5C4×A4
prime radicalD5×SL(2,𝔽3)C2
cosocleQ8×D5C2×C3
LCSG ▷ C5×Q8
UCSC1 ◁ C2

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 40

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#226, #11756704, #11756705, …
maximal normal subgroup ofAGL(1,𝔽5)×#226, #11756704, #11756705, …
maximal quotient ofSL(2,𝔽3)×#170, C2×AGL(1,𝔽5)×SL(2,𝔽3), AGL(1,𝔽5)×Q8⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(480, 965);