C₁₀₉×#1304 #28358558

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#1304

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#95
commutator	C₂³	C₄²×C₁₀₉
Frattini, cosocle	C₂⁵	C₂²×C₁₀₉
prime radical	C₂⁵×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂⁵×C₁₀₉ ◁ G

396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 125 points:

(1,2)(3,9,11,14)(4,7)(5,8)(6,10,13,16)(12,15)(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);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,15,9)(12,16)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7394, C₁₀₉×#7426, C₂×C₁₀₉×#1304, …
maximal normal subgroup of	C₁₀₉×#7394, C₁₀₉×#7426, C₂×C₁₀₉×#1304, …
maximal quotient of	C₁₀₉×#7032, C₁₀₉×#7035, C₁₀₉×#7047, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 36);

C109×#1304 #28358558