C₁₀₉×#3006 #28360260

order	13952
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₀₉×#3006

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 4 generators on 121 points:

(1,2)(3,4)(6,10)(7,8)(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);(5,6)(7,10)(8,12)(9,11);(2,4);(5,7,9,8)(6,10,11,12)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#9347, C₁₀₉×#9365, C₁₀₉×#9368, …
maximal normal subgroup of	C₁₀₉×#9347, C₁₀₉×#9365, C₁₀₉×#9368, …
maximal quotient of	C₁₀₉×#18135, C₁₀₉×#18140, C₁₀₉×#18145, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1738);

C109×#3006 #28360260