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Cayley diagram

D5×C49 #92434

order490
isoclinismisoclinic class of D5
automorphismAut(G) = C2×C3×C7×AGL(1,𝔽5) (order 840) · Inn(G) = D5 · Out(G) = C22×C3×C7
homologyAbelianization (H1) = C2×C49 · Schur multiplier (H2) = C1

Aliases

direct productD5×C49
unambiguous semidirect productC5⋊(C2×C49)

Splittings

Fitting, HallC5×C49 ⋊ C2
derived, SylowC5 ⋊ C2×C49

Subgroups

subgroupG/H
center, hypercenterC49D5
commutator, hypocenterC5C2×C49
FrattiniC7C7×D5
FittingC5×C49C2
socle, cosocleC5×C7C2×C7
prime radicalC7×D5C7
LCSG ▷ C5
UCSC1 ◁ C49

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 245

A minimal faithful permutation representation — 2 generators on 54 points:

(2,3)(4,5)(6,7,9,12,16,21,27,8,10,13,17,22,28,34,11,14,18,23,29,35,40,15,19,24,30,36,41,45,20,25,31,37,42,46,49,26,32,38,43,47,50,52,33,39,44,48,51,53,54);(1,2,4,5,3)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×C49, C49⋊AGL(1,𝔽5), D5×D49, …
maximal normal subgroup ofAGL(1,𝔽5)×C49, C49⋊AGL(1,𝔽5), D5×D49, …
maximal quotient ofDic5×C49, C2×D5×C49, C3×D5×C49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(490, 1);