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C22×ΓL(1,𝔽8)×AGL(1,𝔽7) #72245273

order3528
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC22×ΓL(1,𝔽8)×AGL(1,𝔽7)

Splittings

Fitting, socleC22×C72 ⋊ C2×C32
derived, SylowC72 ⋊ C23×C32
HallΓL(1,𝔽8)2 ⋊ C23
HallC22×C7×D7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC22ΓL(1,𝔽8)×AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C23×C32
FrattiniC1G
Fitting, socleC22×C72C2×C32
LCSG ▷ C72
UCSC1 ◁ C22

394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×AGL(1,𝔽7)2, C2×ΓL(1,𝔽8)×(C22⋊D7)⋊C3, C2×#72222015, …
maximal normal subgroup ofC22×AGL(1,𝔽7)2, C2×ΓL(1,𝔽8)×(C22⋊D7)⋊C3, C2×#72222015, …
maximal quotient ofΓL(1,𝔽8)×#1157⋊C3, C2×ΓL(1,𝔽8)×(C22⋊D7)⋊C3, D4×ΓL(1,𝔽8)×AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.