C₁₀₉×#1298 #28358552

order	13952
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1298

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂²⋊C₄
commutator	C₂×C₄	C₄²×C₁₀₉
Frattini, cosocle	C₂³×C₄	C₂²×C₁₀₉
socle	C₂²×C₁₀₉	#95
prime radical	C₂⁴×C₁₀₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 2 generators on 133 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#7397, C₁₀₉×#7433, C₂×C₁₀₉×#1298, …
maximal normal subgroup of	C₁₀₉×#7397, C₁₀₉×#7433, C₂×C₁₀₉×#1298, …
maximal quotient of	C₁₀₉×#7026, C₁₀₉×#7029, C₁₀₉×#7038, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 30);

C109×#1298 #28358552