Finite Group Databasebeta

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C2×AGL(1,𝔽7)×#1044 #72243134

order8400
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C5

Aliases

direct productC2×AGL(1,𝔽7)×#1044

Splittings

Fitting, socleC2×C52×C7 ⋊ C2×C3×C4
derived, HallC52×C7 ⋊ C22×C3×C4
HallC2×D7×#1044 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ C22×C4
SylowC52 ⋊ C2×C4×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C22×#1044
SylowC7 ⋊ C22×C3×#1044

Subgroups

subgroupG/H
center, hypercenterC2AGL(1,𝔽7)×#1044
commutator, hypocenterC52×C7C22×C3×C4
FrattiniC1G
Fitting, socleC2×C52×C7C2×C3×C4
prime radicalC2×AGL(1,𝔽7)×C5⋊D5C2
cosocleC7×C5⋊D5C23×C3
LCSG ▷ C52×C7
UCSC1 ◁ C2

560 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#87830, C2×AGL(1,𝔽7)×D5⋊AGL(1,𝔽5), AGL(1,𝔽7)×#87832, …
maximal normal subgroup ofAGL(1,𝔽7)×#87830, C2×AGL(1,𝔽7)×D5⋊AGL(1,𝔽5), AGL(1,𝔽7)×#87832, …
maximal quotient ofAGL(1,𝔽7)×C22⋊#1044

GAP

Past the SmallGroups library coverage — no SmallGroups id.