Finite Group Databasebeta

← Prev (#16813187) Next (#16813189) →

C7×D255 #16813188

order3570
isoclinismisoclinic class of D255
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×D255
unambiguous semidirect productC3⋊(C7×D85)
unambiguous semidirect productC5⋊(C7×D51)
unambiguous semidirect productC17⋊(C7×D15)

Splittings

Fitting, socle, HallC3×C5×C7×C17 ⋊ C2
derived, HallC3×C5×C17 ⋊ C2×C7
HallC5×C7×C17 ⋊ D3
HallC3×C7×C17 ⋊ D5
HallC7×C17 ⋊ D15
HallC3×C5×C7 ⋊ D17
HallC5×C17 ⋊ D3×C7
HallC3×C17 ⋊ C7×D5
HallC5×C7 ⋊ D51
HallC3×C7 ⋊ D85
SylowC17 ⋊ C7×D15
HallC3×C5 ⋊ C7×D17
SylowC5 ⋊ C7×D51
SylowC3 ⋊ C7×D85

Subgroups

subgroupG/H
center, hypercenterC7D255
commutator, cosocle, hypocenterC3×C5×C17C2×C7
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2
LCSG ▷ C3×C5×C17
UCSC1 ◁ C7

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 1785

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,3,4,5,6,7)(8,9,10)(11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32);(9,10)(12,15)(13,14)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC7×D1275, C7×C3⋊(C5⋊D85), C5×C7×D255, …
maximal normal subgroup ofC5×C7×D255, C49×D255, C72×D255
maximal quotient ofC7×D1275, C7×C3⋊(C5⋊D85), C5×C7×D255, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3570, 15);