C₁₀₁×#2135 #27167489

order	12928
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₁×#2135

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₀₁	C₂×#99
commutator	C₂×C₄	C₂²×C₄×C₁₀₁
Frattini, cosocle	C₂×Q₈	C₂³×C₁₀₁
prime radical	C₂×#41×C₁₀₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂×#41×C₁₀₁ ◁ G

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₀₁×#12712, C₁₀₁×#12713, C₁₀₁×#12721, …
maximal normal subgroup of	C₁₀₁×#12712, C₁₀₁×#12713, C₁₀₁×#12721, …
maximal quotient of	C₁₀₁×#11434, C₁₀₁×#11450, C₁₀₁×#11481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 867);