C₁₀₁×#1617 #27166971

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1617

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
prime radical	C₂³×C₁₀₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#12221, C₁₀₁×#12222, C₁₀₁×#12223, …
maximal normal subgroup of	C₁₀₁×#12221, C₁₀₁×#12222, C₁₀₁×#12223, …
maximal quotient of	C₁₀₁×#7792, C₁₀₁×#7794, C₁₀₁×#7806, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 349);