C₁₀₁×#2230 #27167584

order	12928
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₁×#2230

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₁	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₁
prime radical	#135×C₁₀₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂×C₄×C₁₀₁ ◁ C₄×C₈×C₁₀₁ ◁ G

126 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#12364, C₁₀₁×#12365, C₁₀₁×#12366, …
maximal normal subgroup of	C₁₀₁×#12364, C₁₀₁×#12365, C₁₀₁×#12366, …
maximal quotient of	C₁₀₁×#12823, C₁₀₁×#12950, C₁₀₁×#13116, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 962);