C₁₀₁×#1953 #27167307

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

Aliases

direct product

C₁₀₁×#1953

Subgroups

	subgroup	G/H
center	C₄×C₁₀₁	#121
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂×C₁₀₁	#389
prime radical	C₂³×C₁₀₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ G

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

normal subgroups only

This group has 184 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:

(1,2,6,10,7,11,5,21)(3,12,14,8,16,23,15,24)(4,17,19,25,20,31,18,9)(13,22,27,30,29,32,28,26)(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);(1,3,6,14,7,16,5,15)(2,8,10,23,11,24,21,12)(4,13,19,27,20,29,18,28)(9,22,17,30,25,32,31,26);(1,4,5,18,7,20,6,19)(2,9,10,17,11,25,21,31)(3,13,15,28,16,29,14,27)(8,22,23,30,24,32,12,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#12942, C₁₀₁×#12947, C₁₀₁×#12949, …
maximal normal subgroup of	C₁₀₁×#12942, C₁₀₁×#12947, C₁₀₁×#12949, …
maximal quotient of	C₁₀₁×#9607, C₁₀₁×#9608, C₁₀₁×#9609, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 685);