C₁₀₉×#1332 #28358586

order	13952
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₀₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₉×#1332

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₉	D₈
commutator	C₈	C₂×C₈×C₁₀₉
Frattini, cosocle	C₄×C₈	C₂²×C₁₀₉
socle, prime radical	C₂²×C₁₀₉	#102
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 13952

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

(1,2,3,5,4,6,7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22)(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);(9,10,13,16)(11,18,20,24)(12,21,19,15)(14,22,23,17)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₉×#1332, C₁₀₉×#11591, C₁₀₉×#11592, …
maximal normal subgroup of	C₂×C₁₀₉×#1332, C₁₀₉×#11591, C₁₀₉×#11592, …
maximal quotient of	C₁₀₉×#7069, C₁₀₉×#7073, C₁₀₉×#7113, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 64);

C109×#1332 #28358586