Finite Group Databasebeta

← Prev (#72222006) Next (#72222008) →

C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72222007

order3528
automorphism?
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productC4×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

FittingC4×C72 ⋊ C2×C32
derived, SylowC72 ⋊ C2×C32×C4
HallΓL(1,𝔽8)2 ⋊ C2×C4
HallC4×C7⋊D7 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC4ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, hypocenterC72C2×C32×C4
FrattiniC2C2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
FittingC4×C72C2×C32
socle, cosocleC2×C72C22×C32
prime radicalC2×ΓL(1,𝔽8)⋊AGL(1,𝔽7)C2
LCSG ▷ C72
UCSC1 ◁ C4

224 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.

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

Full subgroup data →

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), AGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C2×C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), …
maximal normal subgroup ofD4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C2×C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72245135, …
maximal quotient ofAGL(1,𝔽5)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C2×C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.