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AGL(1,𝔽5)×#1828 #72269849

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

Aliases

direct productAGL(1,𝔽5)×#1828

Splittings

FittingC5×#1828 ⋊ C4
SylowC5 ⋊ C4×#1828

Subgroups

subgroupG/H
centerC2AGL(1,𝔽5)×#381
commutatorC23×C5C22×C42
FrattiniC24C23×AGL(1,𝔽5)
FittingC5×#1828C4
socleC2×C5C4×#381
prime radicalC22×D4×D5C23
cosocleC24×D5C24
hypercenter#1828AGL(1,𝔽5)
hypocenterC5C4×#1828
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C24 ◁ #1828

3498 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)(6,9,10,17)(7,12)(8,15,16,21)(11,18)(13,19)(14,20);(6,8,10,16)(7,11)(9,15,17,21)(12,18)(13,14)(19,20);(2,3,4,5);(6,7)(8,14)(9,12,17,19)(10,13)(11,16)(15,18,21,20)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#23996, AGL(1,𝔽5)×#21776
maximal normal subgroup ofAGL(1,𝔽5)×#23996, AGL(1,𝔽5)×#21776

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