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Cayley diagram

C2×D49 #5992

order196
isoclinismisoclinic class of D49
automorphismAut(G) = C2×#72215772 (order 4116) · Inn(G) = D49 · Out(G) = C2×C3×C7
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD98
direct productC2×D49
unambiguous semidirect productC49⋊C22
unambiguous semidirect product(C2×C49)⋊C2

Splittings

FittingC2×C49 ⋊ C2
derived, SylowC49 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2D49
commutator, cosocle, hypocenterC49C22
FrattiniC7C2×D7
FittingC2×C49C2
socleC2×C7D7
LCSG ▷ C49
UCSC1 ◁ C2

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 98

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×D49, D196, C22⋊D49, …
maximal normal subgroup ofC4×D49, D196, C22⋊D49, …
maximal quotient ofDic98, C4×D49, D196, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(196, 3);