C₁₀₉×#1806 #28359060

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₉×#1806

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 121
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#8420, C₁₀₉×#8421, C₁₀₉×#8422, …
maximal normal subgroup of	C₁₀₉×#8420, C₁₀₉×#8421, C₁₀₉×#8422, …
maximal quotient of	C₁₀₉×#8563, C₁₀₉×#8564, C₁₀₉×#8565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 538);

C109×#1806 #28359060