C₁₀₉×#3296 #28360550

order	13952
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₉×#3296

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₉
prime radical	#546×C₁₀₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄×D₄×C₁₀₉ ◁ G

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 133 points:

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

Conjugacy classes & character table

3815 conjugacy classes. The character table is too large to display.

Relationships

maximal quotient of

C₁₀₉×#22414, C₁₀₉×#22486, C₁₀₉×#22570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2028);

C109×#3296 #28360550