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AGL(1,𝔽5)×#2127 #72242837

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

Aliases

direct productAGL(1,𝔽5)×#2127

Splittings

FittingC5×#2127 ⋊ C4
SylowC5 ⋊ C4×#2127

Subgroups

subgroupG/H
centerC2C2×AGL(1,𝔽5)×#99
commutatorC2×C4×C5C22×C42
FrattiniC2×D4C23×AGL(1,𝔽5)
FittingC5×#2127C4
socleC2×C5C2×C4×#99
prime radicalD5×#535C22
cosocleC2×D4×D5C24
hypercenter#2127AGL(1,𝔽5)
hypocenterC5C4×#2127
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #2127

3078 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)(7,12)(9,17)(11,18)(15,21);(2,3,4,5);(6,8)(7,11)(9,15,17,21)(10,16)(12,18)(13,14,19,20);(6,7)(8,14,16,20)(9,13)(10,12)(11,15,18,21)(17,19)

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#12753, AGL(1,𝔽5)×#33551, AGL(1,𝔽5)×#12707
maximal normal subgroup ofAGL(1,𝔽5)×#12753, AGL(1,𝔽5)×#33551, AGL(1,𝔽5)×#12707

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