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C2×C7×D765 #36339284

order21420
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×D765
unambiguous semidirect product(C2×C5)⋊(C7×D153)
unambiguous semidirect product(C2×C9)⋊(C7×D85)
unambiguous semidirect product(C2×C17)⋊(C7×D45)

Splittings

FittingC2×C5×C7×C9×C17 ⋊ C2
derived, HallC5×C9×C17 ⋊ C22×C7
HallC5×C7×C9×C17 ⋊ C22
HallC7×C9×C17 ⋊ C2×D5
HallC5×C7×C17 ⋊ C2×D9
HallC5×C7×C9 ⋊ C2×D17
HallC9×C17 ⋊ C2×C7×D5
HallC7×C17 ⋊ C2×D45
HallC5×C17 ⋊ C2×C7×D9
HallC7×C9 ⋊ C2×D85
HallC5×C9 ⋊ C2×C7×D17
HallC5×C7 ⋊ C2×D153
SylowC17 ⋊ C2×C7×D45
SylowC9 ⋊ C2×C7×D85
SylowC5 ⋊ C2×C7×D153

Subgroups

subgroupG/H
center, hypercenterC2×C7D765
commutator, cosocle, hypocenterC5×C9×C17C22×C7
FrattiniC3C2×C7×D255
FittingC2×C5×C7×C9×C17C2
socleC2×C3×C5×C7×C17D3
LCSG ▷ C5×C9×C17
UCSC1 ◁ C2×C7

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 10710

A minimal faithful permutation representation — 2 generators on 40 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(19,20)(21,22)(24,26)(25,29)(27,28)(30,31)(32,33,34,35,36,37,38);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,21,22,20)(23,31,26,29,28,27,25,24,30)(39,40)

Conjugacy classes & character table

5376 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21420, 116);