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AGL(1,𝔽5)×#3811 #72224563

order2880
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productAGL(1,𝔽5)×#3811

Splittings

FittingC22×C32×C5 ⋊ C22×C4
HallC32×C5 ⋊ C2×C4×D4
SylowC32 ⋊ C2×D4×AGL(1,𝔽5)
SylowC5 ⋊ C4×#3811

Subgroups

subgroupG/H
center, FrattiniC2C2×D32×AGL(1,𝔽5)
commutator, socleC2×C32×C5C23×C4
FittingC22×C32×C5C22×C4
prime radicalD5×#3811C2
cosocleC2×C32×D5C24
hypercenterC22D32×AGL(1,𝔽5)
hypocenterC32×C5C2×C4×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C22

1940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224650, C2×AGL(1,𝔽5)×#3811, D32×D4×AGL(1,𝔽5), …
maximal normal subgroup ofC2×AGL(1,𝔽5)×#3811, D32×D4×AGL(1,𝔽5), AGL(1,𝔽5)×#64306, …
maximal quotient of#72224650, C2×AGL(1,𝔽5)×#3811, AGL(1,𝔽5)×#63930, …
holomorph ofC5×Dic3, C2×C5×D3

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,5,3,4)(11,12)(13,14), (2,3,4,5)(10,11,12), (7,9), (6,7,8,9)(14,15));