C₁₀₉×#3237 #28360491

order	13952
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#3237

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 141 points:

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

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#9374, C₁₀₉×#10043, C₁₀₉×#10045, …
maximal normal subgroup of	C₁₀₉×#9374, C₁₀₉×#10043, C₁₀₉×#10045, …
maximal quotient of	C₁₀₉×#21839, C₁₀₉×#21870, C₁₀₉×#21988, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1969);

C109×#3237 #28360491