C₁₀₉×#1537 #28358791

order	13952
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#1537

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 141
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 141 points:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18)(19,26,28,31)(20,29)(21,24)(22,25)(23,27,32,30)(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);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Conjugacy classes & character table

2834 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₉×#8477, C₁₀₉×#8478, C₁₀₉×#8494, …
maximal normal subgroup of	C₁₀₉×#8477, C₁₀₉×#8478, C₁₀₉×#8494, …
maximal quotient of	C₁₀₉×#7598, C₁₀₉×#7604, C₁₀₉×#7646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 269);

C109×#1537 #28358791