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C7×D1265 #31799320

order17710
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D1265
unambiguous semidirect productC5⋊(C7×D253)
unambiguous semidirect productC11⋊(C7×D115)
unambiguous semidirect productC23⋊(C7×D55)

Splittings

Fitting, socle, HallC5×C7×C11×C23 ⋊ C2
derived, HallC5×C11×C23 ⋊ C2×C7
HallC7×C11×C23 ⋊ D5
HallC5×C7×C23 ⋊ D11
HallC5×C7×C11 ⋊ D23
HallC11×C23 ⋊ C7×D5
HallC7×C23 ⋊ D55
HallC5×C23 ⋊ C7×D11
HallC7×C11 ⋊ D115
HallC5×C11 ⋊ C7×D23
HallC5×C7 ⋊ D253
SylowC23 ⋊ C7×D55
SylowC11 ⋊ C7×D115
SylowC5 ⋊ C7×D253

Subgroups

subgroupG/H
center, hypercenterC7D1265
commutator, cosocle, hypocenterC5×C11×C23C2×C7
FrattiniC1G
Fitting, socleC5×C7×C11×C23C2
LCSG ▷ C5×C11×C23
UCSC1 ◁ C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 8855

Conjugacy classes & character table

4438 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×D23×D55, C7×D11×D115, C7×D5×D253, …
maximal normal subgroup ofC7×D23×D55, C7×D11×D115, C7×D5×D253, …
maximal quotient ofC7×Dic1265, C2×C7×D1265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17710, 14);