C₁₀₉×#1829 #28359083

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#1829

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#381
commutator	C₂³	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	C₂²×D₄×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 1744

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2,4,8,5,6,12,13)(3,9,11,16,10,7,14,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(2,6)(4,12)(7,9)(11,14);(1,3)(2,7,6,9)(4,11)(5,10)(8,15,13,16)(12,14)

Conjugacy classes & character table

3161 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12725, C₁₀₉×#12726, C₂×C₁₀₉×#1829, …
maximal normal subgroup of	C₁₀₉×#12725, C₁₀₉×#12726, C₂×C₁₀₉×#1829, …
maximal quotient of	C₁₀₉×#8734, C₁₀₉×#8736, C₁₀₉×#8751, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 561);

C109×#1829 #28359083