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C22×C22⋊C4×AGL(1,𝔽7) #72228883

order2688
automorphism?
homologyAbelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C211

Aliases

direct productC22×C22⋊C4×AGL(1,𝔽7)

Splittings

FittingC22×C7×C22⋊C4 ⋊ C2×C3
HallC22×D7×C22⋊C4 ⋊ C3
HallΓL(1,𝔽8) ⋊ C23×C22⋊C4
SylowC7 ⋊ C23×C3×C22⋊C4

Subgroups

subgroupG/H
centerC24C22×AGL(1,𝔽7)
commutatorC2×C7C24×C3×C4
FrattiniC22C24×AGL(1,𝔽7)
FittingC22×C7×C22⋊C4C2×C3
socleC24×C7C23×C3
prime radicalC25×AGL(1,𝔽7)C2
cosocleC22×C7C25×C3
hypercenterC22×C22⋊C4AGL(1,𝔽7)
hypocenterC7C23×C3×C22⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C22⋊C4

9264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 224

A minimal faithful permutation representation — 5 generators on 19 points:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC22×#120×AGL(1,𝔽7), AGL(1,𝔽7)×#1898, AGL(1,𝔽7)×#1438, …
maximal normal subgroup ofC22×#120×AGL(1,𝔽7), AGL(1,𝔽7)×#1898, AGL(1,𝔽7)×#1438, …

GAP

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

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