C₄×C₉×C₁₀₉ #16840974

order	3924
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₄×C₂₇ (order 1296) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₄×C₂₇
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₃₉₂₄
abelian	C₄×C₉×C₁₀₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃	C₂×C₃×C₁₀₉
socle, prime radical	C₂×C₃×C₁₀₉	C₂×C₃
LCS	G ▷ C₁
UCS	C₁ ◁ G

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 122
transitive degree	μ_t(G) = 3924

A minimal faithful permutation representation — 1 generator on 122 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122)

Conjugacy classes & character table

3924 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₄×C₅×C₉×C₁₀₉, C₉×Dic₅×C₁₀₉, C₉×AGL(1,𝔽₅)×C₁₀₉, …
maximal normal subgroup of	C₄×C₅×C₉×C₁₀₉, C₄×C₇×C₉×C₁₀₉, C₈×C₉×C₁₀₉, …
maximal quotient of	C₄×C₅×C₉×C₁₀₉, C₉×Dic₅×C₁₀₉, C₉×AGL(1,𝔽₅)×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3924, 1);

C4×C9×C109 #16840974