C₁₀₁×#1785 #27167139

order	12928
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₀₁×#1785

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
prime radical	C₂²×#41×C₁₀₁	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂³×C₄×C₁₀₁ ◁ G

584 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 121
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 121 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#7306, C₁₀₁×#7309, C₁₀₁×#7312, …
maximal normal subgroup of	C₁₀₁×#7306, C₁₀₁×#7309, C₁₀₁×#7312, …
maximal quotient of	C₁₀₁×#8300, C₁₀₁×#8301, C₁₀₁×#8304, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 517);