C₁₀₁×#1664 #27167018

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1664

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₁₀₁
prime radical	C₂×D₄×C₁₀₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ G

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

normal subgroups only

This group has 184 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) = 133
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(19,25)(20,27)(22,30)(23,28)(26,29)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,18,21,24)(19,22,26,31)(20,28,27,23)(25,32,29,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,19,21,26)(18,22,24,31)(20,29,27,25)(23,32,28,30)

Conjugacy classes & character table

2020 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₁×#1664, C₁₀₁×#18987, C₁₀₁×#18990, …
maximal normal subgroup of	C₂×C₁₀₁×#1664, C₁₀₁×#18987, C₁₀₁×#18990, …
maximal quotient of	C₁₀₁×#7764, C₁₀₁×#7765, C₁₀₁×#7766, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 396);