C₁₀₉×#2050 #28359304

order	13952
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#2050

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#394
prime radical	#575×C₁₀₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

266 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7414, C₁₀₉×#7415, C₁₀₉×#12173, …
maximal normal subgroup of	C₁₀₉×#7414, C₁₀₉×#7415, C₁₀₉×#12173, …
maximal quotient of	C₁₀₉×#10456, C₁₀₉×#10457, C₁₀₉×#10458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 782);