C₁₃₁×#32875 #45718759

order	33536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₁×#32875

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃₁	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₁
prime radical	C₂³×D₄×C₁₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

712 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 147
transitive degree	μ_t(G) = 2096

A minimal faithful permutation representation — 4 generators on 147 points:

(2,6)(3,9)(4,12)(7,14)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,2,4,13,5,6,12,8)(3,10,11,7,9,15,16,14);(1,3,4,11,5,9,12,16)(2,7,8,10,6,14,13,15);(7,14)(10,15)

Conjugacy classes & character table

3668 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 25863);