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M16×AGL(1,𝔽13) #72229128

order2496
automorphism?
homologyAbelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C2×C4

Aliases

direct productM16×AGL(1,𝔽13)

Splittings

FittingC13×M16 ⋊ C3×C4
HallM16×#261 ⋊ C3
HallC13⋊C3 ⋊ C4×M16
SylowC13 ⋊ C3×C4×M16

Subgroups

subgroupG/H
center, FrattiniC4C22×AGL(1,𝔽13)
commutator, socleC2×C13C2×C3×C42
FittingC13×M16C3×C4
prime radicalC22×D13⋊C3C2×C4
cosocleC4×D13C23×C3
hypercenterM16AGL(1,𝔽13)
hypocenterC13C3×C4×M16
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C4 ◁ M16

284 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 104

A minimal faithful permutation representation — 3 generators on 21 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/8)×AGL(1,𝔽13), #72229302, #100×AGL(1,𝔽13), …
maximal normal subgroup ofAGL(1,ℤ/8)×AGL(1,𝔽13), #100×AGL(1,𝔽13), #104×AGL(1,𝔽13)
maximal quotient of#72229302, #72259347, #97×AGL(1,𝔽13)

GAP

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

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