C₁₀₁×#1654 #27167008

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1654

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₁
prime radical	#547×C₁₀₁	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ G

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

normal subgroups only

This group has 240 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) = 117
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

2626 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₁×#1654, C₁₀₁×#18503, C₁₀₁×#18517, …
maximal normal subgroup of	C₂×C₁₀₁×#1654, C₁₀₁×#18503, C₁₀₁×#18517, …
maximal quotient of	C₁₀₁×#7770, C₁₀₁×#7772, C₁₀₁×#7778, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 386);