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AGL(1,𝔽5)×#1913 #72242835

order2560
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4

Aliases

direct productAGL(1,𝔽5)×#1913

Splittings

FittingC5×#1913 ⋊ C4
SylowC5 ⋊ C4×#1913

Subgroups

subgroupG/H
centerC2AGL(1,𝔽5)×#386
commutatorC23×C5C22×C42
FrattiniC24C23×AGL(1,𝔽5)
FittingC5×#1913C4
socleC2×C5C4×#386
prime radicalD5×#534C22
cosocleC24×D5C24
hypercenter#1913AGL(1,𝔽5)
hypocenterC5C4×#1913
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C24 ◁ #1913

4438 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)(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20);(2,3,4,5);(6,8)(7,11,13,14)(9,17)(10,16)(12,20,21,19)(15,18);(6,7,10,13)(8,14,16,11)(9,12,18,21)(15,19,17,20)

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#12753, AGL(1,𝔽5)×#23895, AGL(1,𝔽5)×#23996, …
maximal normal subgroup ofAGL(1,𝔽5)×#12753, AGL(1,𝔽5)×#23895, AGL(1,𝔽5)×#23996, …

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