Finite Group Databasebeta

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C22⋊C4×ΓL(1,𝔽8)⋊D7 #72246038

order4704
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC22⋊C4×ΓL(1,𝔽8)⋊D7

Splittings

FittingC72×C22⋊C4 ⋊ C2×C3
HallC22⋊C4×C7⋊D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C22⋊C4
SylowC72 ⋊ C2×C3×C22⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×ΓL(1,𝔽8)⋊D7
commutatorC2×C72C22×C3×C4
FittingC72×C22⋊C4C2×C3
socle, cosocleC22×C72C23×C3
prime radicalC23×ΓL(1,𝔽8)⋊D7C2
hypercenterC22⋊C4ΓL(1,𝔽8)⋊D7
hypocenterC72C2×C3×C22⋊C4
LCSG ▷ C2×C72 ▷ C72
UCSC1 ◁ C22 ◁ C22⋊C4

660 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 168

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#120×ΓL(1,𝔽8)⋊D7, C22⋊C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246116, …
maximal normal subgroup of#120×ΓL(1,𝔽8)⋊D7, C22⋊C4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246116, …
maximal quotient of#99×ΓL(1,𝔽8)⋊D7, #100×ΓL(1,𝔽8)⋊D7, #104×ΓL(1,𝔽8)⋊D7

GAP

Past the SmallGroups library coverage — no SmallGroups id.