C₁₀₁×#1349 #27166703

order	12928
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₀₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₀₁×#1349

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#102
commutator	C₂×C₈	C₂×C₄×C₁₀₁
Frattini, cosocle	C₂²×C₈	C₂²×C₁₀₁
prime radical	#466×C₁₀₁	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂²×C₄×C₁₀₁ ◁ C₂²×C₈×C₁₀₁ ◁ G

114 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 125
transitive degree	μ_t(G) = 6464

A minimal faithful permutation representation — 2 generators on 125 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,19,13,18)(11,20,22,16,23,15,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(1,3)(2,5)(4,8)(6,7)(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23)

Conjugacy classes & character table

2929 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₁×#1349, C₁₀₁×#11709, C₁₀₁×#11710, …
maximal normal subgroup of	C₂×C₁₀₁×#1349, C₁₀₁×#11709, C₁₀₁×#11710, …
maximal quotient of	C₁₀₁×#7080, C₁₀₁×#7083, C₁₀₁×#7084, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 81);