C₁₀₉×#1388 #28358642

order	13952
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1388

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	#102
commutator	C₈	C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#340
prime radical	C₂³×C₁₀₉	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 2 generators on 141 points:

(1,2,5,11,8,13,7,24)(3,14,16,26,19,28,18,32)(4,20,22,29,23,12,6,10)(9,17,25,21,27,31,30,15)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,6,17,7,18,23,15,8,19,22,31,5,16,4,21)(2,9,12,26,11,25,10,28,13,27,20,32,24,30,29,14)

Conjugacy classes & character table

3488 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₉×#1388, C₁₀₉×#12646, C₁₀₉×#12651, …
maximal normal subgroup of	C₂×C₁₀₉×#1388, C₁₀₉×#12646, C₁₀₉×#12651, …
maximal quotient of	C₁₀₉×#7158, C₁₀₉×#7176, C₁₀₉×#7177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 120);