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D616 #11927682

order1232
automorphismAut(G) order 147840
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD616

Splittings

FittingC7×C8×C11 ⋊ C2
HallC7×C11 ⋊ D8
SylowC11 ⋊ D56
SylowC7 ⋊ D88

Subgroups

subgroupG/H
centerC2D308
commutator, cosocleC4×C7×C11C22
FrattiniC4C2×D77
FittingC7×C8×C11C2
socleC2×C7×C11D4
hypercenterC8D77
hypocenterC7×C11D8
LCSG ▷ C4×C7×C11 ▷ C2×C7×C11 ▷ C7×C11
UCSC1 ◁ C2 ◁ C4 ◁ C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 616

A minimal faithful permutation representation — 2 generators on 26 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,26)(18,25)(19,24)(20,23)(21,22)

Polycyclic presentation:

2^4,7,11|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^10

Conjugacy classes & character table

311 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD1232, D3080, D1848, …
maximal normal subgroup ofD1232
maximal quotient ofD1232, Dic616, D3080, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1232, 65);