C₁₀₉×#1905 #28359159

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#1905

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#386
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

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

normal subgroups only

This group has 310 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) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2,6,10,7,12,5,11)(3,13,16,23,17,26,15,8)(4,18,20,9,21,24,19,25)(14,22,29,31,30,32,28,27)(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)(2,8,12,23)(4,14)(5,15)(6,16)(7,17)(9,22,25,32)(10,13,11,26)(18,27,24,31)(19,28)(20,29)(21,30);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,23,32)(10,18,11,24)(13,27,26,31)(15,29,16,28)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1905, C₁₀₉×#20998, C₁₀₉×#22284, …
maximal normal subgroup of	C₂×C₁₀₉×#1905, C₁₀₉×#20998, C₁₀₉×#22284, …
maximal quotient of	C₁₀₉×#9088, C₁₀₉×#9090, C₁₀₉×#9092, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 637);