C₁₀₉×#1615 #28358869

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1615

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	#121×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Conjugacy classes & character table

2507 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₉×#1615, C₁₀₉×#18107, C₁₀₉×#18127, …
maximal normal subgroup of	C₂×C₁₀₉×#1615, C₁₀₉×#18107, C₁₀₉×#18127, …
maximal quotient of	C₁₀₉×#7787, C₁₀₉×#7788, C₁₀₉×#7798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 347);