C₁₀₁×#1289 #27166643

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₁×#1289

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#95
commutator	C₂×C₄	C₄²×C₁₀₁
Frattini, cosocle	C₂³×C₄	C₂²×C₁₀₁
prime radical	C₂⁴×C₁₀₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 3232

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2)(3,7,8,5)(4,6)(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);(1,3)(2,5)(4,8)(6,7)(9,10,12,15,13,16,11,14)

Conjugacy classes & character table

3232 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#7076, C₁₀₁×#7077, C₁₀₁×#7078, …
maximal normal subgroup of	C₁₀₁×#7076, C₁₀₁×#7077, C₁₀₁×#7078, …
maximal quotient of	C₁₀₁×#7020, C₁₀₁×#7024, C₁₀₁×#7035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 21);