C₁₀₁×#1460 #27166814

order	12928
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₀₁×#1460

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₁	C₂²⋊C₄
commutator	C₂²	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₁
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₀₁	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

312 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 117
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 3 generators on 117 points:

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

Conjugacy classes & character table

4444 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₀₁×#7204, C₁₀₁×#7285, C₈×#100×C₁₀₁, …
maximal normal subgroup of	C₁₀₁×#7204, C₁₀₁×#7285, C₈×#100×C₁₀₁, …
maximal quotient of	C₁₀₁×#7567, C₁₀₁×#7569, C₁₀₁×#7571, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 192);