C₁₀₁×#1976 #27167330

order	12928
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₁×#1976

Subgroups

	subgroup	G/H
center	C₄×C₁₀₁	#121
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂×C₁₀₁	#390
prime radical	C₂×#135×C₁₀₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 108 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)(3,7,12,21)(4,14)(5,17)(6,9)(8,15)(10,18)(11,23,24,32)(13,27,26,20)(16,28)(19,30,31,25)(22,29)(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)(2,7,9,21)(4,11)(5,13)(6,12)(8,19,22,31)(10,20,17,27)(14,23,28,32)(15,25)(16,24)(18,26)(29,30);(1,4,5,15,6,16,18,29)(2,8,10,14,9,22,17,28)(3,11,13,25,12,24,26,30)(7,19,20,23,21,31,27,32)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#13016, C₁₀₁×#13021, C₁₀₁×#13030, …
maximal normal subgroup of	C₁₀₁×#13016, C₁₀₁×#13021, C₁₀₁×#13030, …
maximal quotient of	C₁₀₁×#9667, C₁₀₁×#9668, C₁₀₁×#9669, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 708);