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Cayley diagram

D123 #6930

order246
isoclinismisoclinic class of D123
automorphismAut(G) = D3×AGL(1,𝔽41) (order 9840) · Inn(G) = G · Out(G) = C5×C8
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD123
unambiguous semidirect productC3⋊D41
unambiguous semidirect productC41⋊D3

Splittings

Fitting, socle, derived, HallC3×C41 ⋊ C2
SylowC41 ⋊ D3
SylowC3 ⋊ D41

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C41C2
LCSG ▷ C3×C41
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 123

A minimal faithful permutation representation — 2 generators on 44 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(2,3)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)

Polycyclic presentation:

2,3,41|b^a=b^2,c^a=c^40

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D41, C2×D123, D369, …
maximal normal subgroup ofD3×D41, C2×D123, C3×D123, …
maximal quotient ofDic123, C2×D123, D369, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(246, 3);