C₁₀₉×#2078 #28359332

order	13952
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₉×#2078

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	C₂×#41×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

198 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 6976

A minimal faithful permutation representation — 3 generators on 133 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₉×#12284, C₁₀₉×#12285, C₁₀₉×#12286, …
maximal normal subgroup of	C₁₀₉×#12284, C₁₀₉×#12285, C₁₀₉×#12286, …
maximal quotient of	C₁₀₉×#10615, C₁₀₉×#10616, C₁₀₉×#10617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13952, 810);