C₁₀₉×#3280 #28360534

order	13952
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₉×#3280

Subgroups

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

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 129 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#33830, C₁₀₉×#33832, C₁₀₉×#33833, …
maximal normal subgroup of	C₁₀₉×#33830, C₁₀₉×#33832, C₁₀₉×#33833, …
maximal quotient of	C₁₀₉×#19771, C₁₀₉×#19772, C₁₀₉×#19773, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2012);

C109×#3280 #28360534