C₁₀₁×#2082 #27167436

order	12928
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#2082

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₁	#397
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
prime radical	C₂³×C₁₀₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 3232

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

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

Conjugacy classes & character table

2323 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₁×#2082, C₁₀₁×#23652, C₁₀₁×#24987, …
maximal normal subgroup of	C₂×C₁₀₁×#2082, C₁₀₁×#23652, C₁₀₁×#24987, …
maximal quotient of	C₁₀₁×#10632, C₁₀₁×#10633, C₁₀₁×#10635, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 814);