C₁₀₁×#20319 #38802470

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

Aliases

direct product

C₁₀₁×#20319

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₀₁
prime radical	#493×C₁₀₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₄²×C₁₀₁ ◁ G

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

normal subgroups only

This group has 578 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) = 12928

A minimal faithful permutation representation — 4 generators on 133 points:

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

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, 13307);