C₁₀₁×#2214 #27167568

order	12928
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₀₁×#2214

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₁	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₀₁
prime radical	#135×C₁₀₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂×C₄×C₁₀₁ ◁ C₄×C₈×C₁₀₁ ◁ G

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Conjugacy classes & character table

2323 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#12312, C₁₀₁×#12313, C₁₀₁×#12315, …
maximal normal subgroup of	C₁₀₁×#12312, C₁₀₁×#12313, C₁₀₁×#12315, …
maximal quotient of	C₁₀₁×#12422, C₁₀₁×#12423, C₁₀₁×#12426, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 946);