C₁₀₉×#1647 #28358901

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#1647

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	#547×C₁₀₉	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ G

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 133
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 133 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1647, C₁₀₉×#18503, C₁₀₉×#18516, …
maximal normal subgroup of	C₂×C₁₀₉×#1647, C₁₀₉×#18503, C₁₀₉×#18516, …
maximal quotient of	C₁₀₉×#7751, C₁₀₉×#7752, C₁₀₉×#7763, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 379);