C₁₀₉×#1765 #28359019

order	13952
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₀₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₉×#1765

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator	C₄	C₂×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂²×C₈×C₁₀₉ ◁ G

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 3 generators on 129 points:

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

Conjugacy classes & character table

4796 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#12618, C₁₀₉×#13127, C₂×C₁₀₉×#1765, …
maximal normal subgroup of	C₁₀₉×#12618, C₁₀₉×#13127, C₂×C₁₀₉×#1765, …
maximal quotient of	C₁₀₉×#8185, C₁₀₉×#8186, C₁₀₉×#8188, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 497);

C109×#1765 #28359019