C₁₀₁×#10042 #38792193

order	25856
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₀₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₁×#10042

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
prime radical	C₁₀₁×#3100	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂×C₄²×C₁₀₁ ◁ G

498 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,22,26)(19,27,30,31)(20,25,21,24)(23,28,32,29)(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);(6,15)(9,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(18,24)(19,28)(20,21)(25,26)(27,31)(29,30)

Conjugacy classes & character table

3131 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25856, 3030);