C₁₀₉×#1409 #28358663

order	13952
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₉×#1409

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#353
commutator	C₄²	C₂×C₄×C₁₀₉
Frattini, cosocle	#127	C₂²×C₁₀₉
prime radical	#127×C₁₀₉	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ #127×C₁₀₉ ◁ G

110 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)(3,10,11,6)(4,8,14,7)(5,9)(12,15,13,16)(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);(1,3,5,12)(2,6,9,16)(4,13,14,11)(7,15,8,10)

Conjugacy classes & character table

1853 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7519, C₁₀₉×#7520, C₁₀₉×#7527, …
maximal normal subgroup of	C₁₀₉×#7519, C₁₀₉×#7520, C₁₀₉×#7527, …
maximal quotient of	C₁₀₉×#7355, C₁₀₉×#7371, C₁₀₉×#7403, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 141);

C109×#1409 #28358663