C₁₀₉×#1392 #28358646

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

Aliases

direct product

C₁₀₉×#1392

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	#99
commutator	C₂³	C₄²×C₁₀₉
Frattini, cosocle	C₂×#41	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#342
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂×#41×C₁₀₉ ◁ G

158 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7400, C₁₀₉×#7401, C₁₀₉×#7407, …
maximal normal subgroup of	C₁₀₉×#7400, C₁₀₉×#7401, C₁₀₉×#7407, …
maximal quotient of	C₁₀₉×#7188, C₁₀₉×#7203, C₁₀₉×#7216, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 124);