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Cayley diagram

D49 #1026

order98
isoclinismisoclinic class of D49
automorphismAut(G) = #72215772 (order 2058) · Inn(G) = G · Out(G) = C3×C7
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD49
unambiguous semidirect productC49⋊C2

Splittings

Fitting, derived, SylowC49 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC49C2
Frattini, socleC7D7
LCSG ▷ C49
UCSC1

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 49
transitive degreeμt(G) = 49

A minimal faithful permutation representation — 2 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)

Polycyclic presentation:

2,7^2|b^7=c,b^a=b^6c^6,c^a=c^6

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D49, D49⋊C3, C3×D49, …
maximal normal subgroup ofC2×D49, D49⋊C3, C3×D49, …
maximal quotient ofDic49, C2×D49, C3×D49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(98, 1);