C₃×C₄×C₅×C₁₁×C₄₉ #44568299

order	32340
nilpotency class	1
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₂₃₄₀
abelian	C₃×C₄×C₅×C₁₁×C₄₉
multiplicative	(ℤ/32341)ˣ, (ℤ/64682)ˣ

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₇	C₂×C₃×C₅×C₇×C₁₁
socle, prime radical	C₂×C₃×C₅×C₇×C₁₁	C₂×C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 32340

A minimal faithful permutation representation — 1 generator on 72 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61)(62,63,64,65,66,67,68,69,70,71,72)

Conjugacy classes & character table

32340 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	Dic₁₆₁₇₀, D₃₂₃₄₀
maximal subgroup of	Dic₁₆₁₇₀, D₃₂₃₄₀
maximal normal subgroup of	Dic₁₆₁₇₀, D₃₂₃₄₀
automorphism group of	C₃₂₃₄₁
outer automorphism group of	C₃₂₃₄₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 1);

C3×C4×C5×C11×C49 #44568299