C₁₀₉×#1881 #28359135

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

Aliases

direct product

C₁₀₉×#1881

Subgroups

	subgroup	G/H
center	C₄×C₁₀₉	#120
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#386
prime radical	#575×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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)(3,12,15,23)(4,16)(5,10)(6,20)(7,11)(8,14,24,19)(9,17)(13,28,31,22)(18,25)(21,26)(27,29,32,30)(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);(1,3,5,19,7,15,6,14)(2,8,10,23,11,24,20,12)(4,13,17,30,18,31,21,29)(9,22,25,32,26,28,16,27);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(23,32)(24,28)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1881, C₁₀₉×#20939, C₁₀₉×#21477, …
maximal normal subgroup of	C₂×C₁₀₉×#1881, C₁₀₉×#20939, C₁₀₉×#21477, …
maximal quotient of	C₁₀₉×#8953, C₁₀₉×#8954, C₁₀₉×#8955, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 613);