C₁₀₁×#8801 #38790952

order	25856
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₀₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₀₁×#8801

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₀₁	#117
commutator	C₂×C₄	C₂²×C₈×C₁₀₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₁
socle	C₂²×C₁₀₁	#383
prime radical	C₂³×C₁₀₁	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₁ ◁ C₂×#97×C₁₀₁ ◁ G

306 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 6464

A minimal faithful permutation representation — 3 generators on 125 points:

(1,2,4,6,3,5,7,8)(9,10)(11,19,22,23)(12,15)(13,17)(14,18)(16,21,24,20)(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);(10,15,18,17)(16,23,24,19);(2,5)(6,8)(9,11,12,20,14,22,13,21)(10,16,17,19,18,24,15,23)

Conjugacy classes & character table

6464 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25856, 1789);