C₁₀₁×#3131 #27168485

order	12928
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₀₁×#3131

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₀₁
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	#532×C₁₀₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

400 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 6464

A minimal faithful permutation representation — 4 generators on 125 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,18)(14,19)(15,20)(21,24)(22,23)(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);(1,3,4,7)(2,5,6,8)(9,11,13,21,15,22,14,12)(10,16,19,23,20,24,18,17);(2,6)(5,8)(9,12,14,22,15,21,13,11)(10,17,18,24,20,23,19,16);(9,13,15,14)(10,18,20,19)(11,21,22,12)(16,17,24,23)

Conjugacy classes & character table

4444 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#9067, C₁₀₁×#9070, C₁₀₁×#9751, …
maximal normal subgroup of	C₁₀₁×#9067, C₁₀₁×#9070, C₁₀₁×#9751, …
maximal quotient of	C₁₀₁×#20760, C₁₀₁×#21083, C₁₀₁×#21195, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 1863);