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C7×C3⋊(C7⋊#595) #23756243

order9996
automorphism?
homologyAbelianization (H1) = C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×C3⋊(C7⋊#595)
unambiguous semidirect productC3⋊(C7×C7⋊#595)
unambiguous semidirect productC7⋊(C7×C3⋊#595)

Splittings

Fitting, socle, HallC3×C72×C17 ⋊ C4
derivedC3×C7×C17 ⋊ C4×C7
HallC72×C17 ⋊ Dic3
HallC3×C72 ⋊ #595
HallC3×C17 ⋊ C7×Dic7
SylowC72 ⋊ C3⋊#595
SylowC17 ⋊ C7×Dic21
SylowC3 ⋊ C7×C7⋊#595

Subgroups

subgroupG/H
center, hypercenterC7C3⋊(C7⋊#595)
commutator, hypocenterC3×C7×C17C4×C7
FrattiniC1G
Fitting, socleC3×C72×C17C4
prime radicalC3×C72×D17C2
cosocleC3×C7×D17C2×C7
LCSG ▷ C3×C7×C17
UCSC1 ◁ C7

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

756 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#44472554, C7×C7⋊(C9⋊#595), C3⋊#23756203, …
maximal normal subgroup ofC7×C3⋊(ΓL(1,𝔽8)⋊#595), ΓL(1,𝔽8)×C3⋊(C7⋊#595), C3⋊#23756186, …
maximal quotient ofC7×C7⋊(C9⋊#595), C3×C7×C3⋊(C7⋊#595), C7×C3⋊(C3⋊(C7⋊#595))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 146);