C₁₀₁×#7308 #38789459

order	25856
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₀₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₀₁×#7308

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₀₁	#342
commutator	C₂²×C₄	C₄²×C₁₀₁
Frattini, cosocle	C₂³×Q₈	C₂²×C₁₀₁
prime radical	C₂⁴×C₁₀₁	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₁ ◁ C₂⁴×C₁₀₁ ◁ C₂³×Q₈×C₁₀₁ ◁ G

518 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 125
transitive degree	μ_t(G) = 3232

A minimal faithful permutation representation — 2 generators on 125 points:

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

Conjugacy classes & character table

3434 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25856, 296);