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Cayley diagram

C5×C49 #6926

order245
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×C4×C7 (order 168) · Inn(G) = C1 · Out(G) = C2×C3×C4×C7
holomorphG ⋊ Aut(G) = AGL(1,𝔽5)×#72215772
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC245
abelianC5×C49

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC7C5×C7
socle, prime radicalC5×C7C7
LCSG ▷ C1
UCSC1 ◁ G

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 1 generator on 54 points:

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

Relationships

center ofC5×D3×C49, C5×D5×C49, C5×A4×C49, …
derived subgroup ofD245, Dic245, C49⋊AGL(1,𝔽5), …
Frattini subgroup ofC2×D8575, D25725
Fitting subgroup ofD5×C49, C5×D49, D245, …
maximal subgroup ofD5×C49, C5×D49, D245, …
maximal normal subgroup ofD5×C49, C5×D49, D245, …
maximal quotient ofC2×C5×C49, C3×C5×C49, C25×C49, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(245, 1);