C₁₀₁×#1572 #27166926

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

Aliases

direct product

C₁₀₁×#1572

Subgroups

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

150 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 6464

A minimal faithful permutation representation — 3 generators on 133 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(3,9)(6,12)(10,15)(13,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)

Conjugacy classes & character table

3838 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₁₀₁×#1572, C₁₀₁×#17803, C₁₀₁×#17808, …
maximal normal subgroup of	C₂×C₁₀₁×#1572, C₁₀₁×#17803, C₁₀₁×#17808, …
maximal quotient of	C₁₀₁×#7878, C₁₀₁×#8277, C₁₀₁×#8278, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12928, 304);