C₁₀₉×#3287 #28360541

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×#3287

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

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 6976

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#12317, C₁₀₉×#12321, C₁₀₉×#12369, …
maximal normal subgroup of	C₁₀₉×#12317, C₁₀₉×#12321, C₁₀₉×#12369, …
maximal quotient of	C₁₀₉×#19867, C₁₀₉×#19868, C₁₀₉×#19869, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 2019);

C109×#3287 #28360541