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Cayley diagram

Dic49 #5990

order196
isoclinismisoclinic class of D49
automorphismAut(G) = C2×#72215772 (order 4116) · Inn(G) = D49 · Out(G) = C2×C3×C7
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

dicyclicDic49
unambiguous semidirect productC49⋊C4

Splittings

derived, SylowC49 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2D49
commutator, hypocenterC49C4
Frattini, socle, prime radicalC2×C7D7
Fitting, cosocleC2×C49C2
LCSG ▷ C49
UCSC1 ◁ C2

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 196

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(1,3,2,4)(6,53)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofDic98, C4×D49, C2×Dic49, …
maximal normal subgroup ofDic98, C4×D49, C2×Dic49, …
maximal quotient ofC49⋊C8, C2×Dic49, C3×Dic49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(196, 1);