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Cayley diagram

C22×C49 #5993

order196
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×D3×C7 (order 252) · Inn(G) = C1 · Out(G) = C2×C3×D3×C7
holomorphG ⋊ Aut(G) = S4×#72215772
homologyAbelianization (H1) = G · Schur multiplier (H2) = C2

Aliases

abelianC22×C49

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC7C22×C7
socle, prime radicalC22×C7C7
LCSG ▷ C1
UCSC1 ◁ G

15 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)(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);(3,4)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

center ofC22⋊C4×C49, C4⋊C4×C49, C2×D4×C49, …
derived subgroup ofC22⋊(C49⋊C3), A4×D49, D49⋊A4, …
Fitting subgroup ofC2×Dic49, C22⋊D49, C22×D49, …
maximal subgroup ofC2×Dic49, C22⋊D49, C2×C4×C49, …
maximal normal subgroup ofC2×Dic49, C22⋊D49, C2×C4×C49, …
maximal quotient ofC2×C4×C49, D4×C49, Q8×C49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(196, 4);