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D420 #11719826

order840
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD420
unambiguous semidirect productC4⋊D105

Splittings

FittingC3×C4×C5×C7 ⋊ C2
HallC3×C5×C7 ⋊ D4
HallC5×C7 ⋊ D12
HallC3×C7 ⋊ D20
HallC3×C5 ⋊ D28
SylowC7 ⋊ D60
SylowC5 ⋊ D84
SylowC3 ⋊ D140

Subgroups

subgroupG/H
center, FrattiniC2C2×D105
commutator, socle, cosocleC2×C3×C5×C7C22
FittingC3×C4×C5×C7C2
hypercenterC4D105
hypocenterC3×C5×C7D4
LCSG ▷ C2×C3×C5×C7 ▷ C3×C5×C7
UCSC1 ◁ C2 ◁ C4

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 420

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19);(2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)

Polycyclic presentation:

2^3,3,5,7|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^6

Conjugacy classes & character table

213 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13132129, #13132130, #13132134, …
maximal normal subgroup of#13132129, #13132130, #13132134, …
maximal quotient of#13132355, D840, Dic420, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 128);