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AGL(1,𝔽5)×#1504 #72242863

order2560
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productAGL(1,𝔽5)×#1504

Splittings

FittingC5×#1504 ⋊ C4
SylowC5 ⋊ C4×#1504

Subgroups

subgroupG/H
centerC22C2×C22⋊C4×AGL(1,𝔽5)
commutatorC2×C4×C5C22×C42
FrattiniC22×C4C23×AGL(1,𝔽5)
FittingC5×#1504C4
socleC22×C5C2×C4×C22⋊C4
prime radicalD5×#525C22
cosocleC22×C4×D5C24
hypercenter#1504AGL(1,𝔽5)
hypocenterC5C4×#1504
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1504

4104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 80

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2,5,3,4)(15,19)(18,20);(2,3,4,5);(6,8)(7,10)(9,13)(11,12)(14,15,17,19)(16,20,21,18);(6,7)(8,12,13,10)(9,11)(15,18,19,20)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#8453
maximal normal subgroup ofAGL(1,𝔽5)×#8453

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)(15,19)(18,20), (2,3,4,5), (6,8)(7,10)(9,13)(11,12)(14,15,17,19)(16,20,21,18), (6,7)(8,12,13,10)(9,11)(15,18,19,20));