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Cayley diagram

C4×C49 #5991

order196
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C22×C3×C7 (order 84) · Inn(G) = C1 · Out(G) = C22×C3×C7
holomorphG ⋊ Aut(G) = D4×#72215772
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC196
abelianC4×C49
multiplicative(ℤ/197)ˣ, (ℤ/394)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, socle, prime radical, cosocleC2×C7C2×C7
LCSG ▷ C1
UCSC1 ◁ G

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator 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)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

center ofM16×C49, #41×C49, C3⋊C8×C49, …
derived subgroup ofDic196, #11715986, D392, …
Frattini subgroup ofD13720, Dic6860, Dic15092, …
Fitting subgroup ofC49⋊C8, Dic98, C4×D49, …
maximal subgroup ofC49⋊C8, C8×C49, Dic98, …
maximal normal subgroup ofC49⋊C8, C8×C49, Dic98, …
maximal quotient ofC8×C49, C2×C4×C49, Dic3×C49, …
automorphism group ofC197, C2×C197
outer automorphism group ofC197, C2×C197

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(196, 2);