C₁₀₉×#1658 #28358912

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1658

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	#127×C₁₀₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ G

182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 6976

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

(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,2)(3,8)(4,12)(5,7)(6,9)(10,15)(11,16)(13,14)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)

Conjugacy classes & character table

2180 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7524, C₂×C₁₀₉×#1658, C₁₀₉×#18990, …
maximal normal subgroup of	C₁₀₉×#7524, C₂×C₁₀₉×#1658, C₁₀₉×#18990, …
maximal quotient of	C₁₀₉×#7739, C₁₀₉×#7744, C₁₀₉×#7827, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 390);

C109×#1658 #28358912