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C4×(C22⋊D7)⋊C3 #10620764

order672
isoclinismisoclinic class of ΓL(1,𝔽8)⋊Q8
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC4×(C22⋊D7)⋊C3
unambiguous semidirect product(C4×C22⋊D7)⋊C3

Splittings

FittingC22×C4×C7 ⋊ C2×C3
HallC4×C22⋊D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C4×D4
SylowC7 ⋊ C3×C4×D4

Subgroups

subgroupG/H
centerC2×C4C2×AGL(1,𝔽7)
commutatorC2×C7C22×C3×C4
FrattiniC22C22×AGL(1,𝔽7)
FittingC22×C4×C7C2×C3
socle, cosocleC22×C7C23×C3
prime radicalC2×(C22⋊D7)⋊C3C2
hypercenterC22×C4AGL(1,𝔽7)
hypocenterC7C3×C4×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4

188 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 112

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89501⋊C3, ((C22⋊D7)⋊C3)⋊C8, #89500⋊C3, …
maximal normal subgroup of#89501⋊C3, ((C22⋊D7)⋊C3)⋊C8, #89500⋊C3, …
maximal quotient ofC4×(ΓL(1,𝔽8)⋊C4)⋊C4, #89705⋊C3, C4×(C2×AGL(1,𝔽7))⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(672, 346);