C₁₃₁×#11902 #45697786

order	33536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃₁×#11902

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃₁
prime radical	C₂²×D₄×C₁₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄×D₄×C₁₃₁ ◁ G

452 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 155
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 155 points:

(2,6)(3,4)(7,8)(9,10)(11,17)(12,20,22,16)(13,15)(14,19,23,21)(18,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155);(1,2,5,8)(3,6,4,7)(9,11)(10,14,15,23)(12,21)(13,18)(16,24,20,17)(19,22);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(12,22)(14,23)

Conjugacy classes & character table

4847 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 4890);