C₁₀₁×#20302 #38802453

order	25856
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₀₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₀₁×#20302

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ G

690 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 198 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 — 4 generators on 133 points:

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

Conjugacy classes & character table

3737 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25856, 13290);