C₁₀₁×#1288 #27166642

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#1288

Subgroups

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

152 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 125
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 2 generators on 125 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₁₀₁×#1288, C₁₀₁×#7972, C₁₀₁×#7973, …
maximal normal subgroup of	C₂×C₁₀₁×#1288, C₁₀₁×#7972, C₁₀₁×#7973, …
maximal quotient of	C₁₀₁×#7019, C₁₀₁×#7023, C₁₀₁×#7024, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 20);