C₁₀₁×#2127 #27167481

order	12928
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₀₁×#2127

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₁	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂×D₄	C₂³×C₁₀₁
prime radical	#535×C₁₀₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂³×C₁₀₁ ◁ C₂²×D₄×C₁₀₁ ◁ G

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

normal subgroups only

This group has 286 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) = 1616

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#12707, C₁₀₁×#12708, C₁₀₁×#12727, …
maximal normal subgroup of	C₁₀₁×#12707, C₁₀₁×#12708, C₁₀₁×#12727, …
maximal quotient of	C₁₀₁×#11432, C₁₀₁×#11448, C₁₀₁×#11479, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 859);