C₁₀₉×#3531 #28360785

order	13952
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₁₀₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₁₀₉×#3531

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
socle	C₂²×C₁₀₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ G

930 conjugacy classes of subgroups; only the 760 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 760 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) = 6976

A minimal faithful permutation representation — 5 generators on 141 points:

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

Conjugacy classes & character table

4142 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8505, C₁₀₉×#18990, C₁₀₉×#18991, …
maximal normal subgroup of	C₁₀₉×#8505, C₁₀₉×#18990, C₁₀₉×#18991, …
maximal quotient of	C₁₀₉×#34348, C₁₀₉×#34350, C₄×#562×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2263);