C₁₀₉×#1411 #28358665

order	13952
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₀₉×#1411

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₉	#353
commutator	C₄²	C₂×C₄×C₁₀₉
Frattini, cosocle	C₄⋊Q₈	C₂²×C₁₀₉
prime radical	C₂²×C₁₀₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₄⋊Q₈×C₁₀₉ ◁ G

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 3488

A minimal faithful permutation representation — 2 generators on 141 points:

(1,2,4,9,6,11,18,28)(3,12,13,26,15,23,30,31)(5,10,20,27,21,8,17,25)(7,16,22,19,24,29,32,14)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141);(1,3,6,15)(2,7,11,24)(4,16,18,29)(5,19,21,14)(8,23,10,12)(9,26,28,31)(13,17,30,20)(22,27,32,25)

Conjugacy classes & character table

1853 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#7521, C₁₀₉×#7522, C₁₀₉×#7531, …
maximal normal subgroup of	C₁₀₉×#7521, C₁₀₉×#7522, C₁₀₉×#7531, …
maximal quotient of	C₁₀₉×#7355, C₁₀₉×#7370, C₁₀₉×#7405, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 143);

C109×#1411 #28358665