C₁₀₉×#1914 #28359168

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#1914

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#386
commutator	C₂³	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	#535×C₁₀₉	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

340 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#12755, C₁₀₉×#12756, C₁₀₉×#12769, …
maximal normal subgroup of	C₁₀₉×#12755, C₁₀₉×#12756, C₁₀₉×#12769, …
maximal quotient of	C₁₀₉×#9107, C₁₀₉×#9109, C₁₀₉×#9187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 646);