C₃×C₉×C₄₉ #13050489

order	1323
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 4536
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

abelian

C₃×C₉×C₄₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₇	C₃²×C₇
socle, prime radical	C₃²×C₇	C₃×C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 1323

A minimal faithful permutation representation — 2 generators on 61 points:

(1,2,3,4,5,6,7,8,9)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(10,11,12)

Conjugacy classes & character table

1323 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃×C₉×C₄₉, C₃×C₂₇×C₄₉, C₉²×C₄₉, …
maximal normal subgroup of	C₂×C₃×C₉×C₄₉, C₃×C₂₇×C₄₉, C₉²×C₄₉, …
maximal quotient of	C₂×C₃×C₉×C₄₉, C₃×C₂₇×C₄₉, C₉²×C₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1323, 9);

C3×C9×C49 #13050489