C₁₀₉×#3246 #28360500

order	13952
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#3246

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#532×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

392 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 4 generators on 125 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#10027, C₁₀₉×#10029, C₁₀₉×#10193, …
maximal normal subgroup of	C₁₀₉×#10027, C₁₀₉×#10029, C₁₀₉×#10193, …
maximal quotient of	C₁₀₉×#21884, C₁₀₉×#21885, C₁₀₉×#21959, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 1978);

C109×#3246 #28360500