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C7×C7⋊(C52⋊C3) #16814727

order3675
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C5

Aliases

direct productC7×C7⋊(C52⋊C3)
unambiguous semidirect productC7⋊(C7×C52⋊C3)
unambiguous semidirect productC52⋊(C7×ΓL(1,𝔽8))

Splittings

Fitting, socle, HallC52×C72 ⋊ C3
derivedC52×C7 ⋊ C3×C7
SylowC72 ⋊ C52⋊C3
SylowC52 ⋊ C7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC7C7⋊(C52⋊C3)
commutator, cosocle, hypocenterC52×C7C3×C7
FrattiniC1G
Fitting, socleC52×C72C3
LCSG ▷ C52×C7
UCSC1 ◁ C7

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 105

Conjugacy classes & character table

427 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×ΓL(1,𝔽8)×C52⋊C3, ΓL(1,𝔽8)×C7⋊(C52⋊C3), #3861⋊(C52⋊C3), …
maximal normal subgroup ofC7×ΓL(1,𝔽8)×C52⋊C3, ΓL(1,𝔽8)×C7⋊(C52⋊C3), #3861⋊(C52⋊C3), …
maximal quotient ofC7×C7⋊(C52⋊C9), C3×C7×C7⋊(C52⋊C3), C7×C22⋊(C7⋊(C52⋊C3))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3675, 16);