C₁₃₁×#7471 #45693355

order	33536
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃₁ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₁×#7471

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	D₁₆
commutator	C₁₆	C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₃₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ C₂²×C₁₆×C₁₃₁ ◁ G

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 171
transitive degree	μ_t(G) = 33536

A minimal faithful permutation representation — 2 generators on 171 points:

(1,2)(3,4)(5,6,7,8)(10,16)(11,12)(13,21)(14,27)(17,29)(18,33)(19,32)(20,31)(22,38)(23,26)(24,25)(28,37)(30,35)(34,39)(36,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(2,4)(9,10,12,17,13,18,24,32,27,35,38,40,37,39,23,31,15,20,26,34,28,36,22,30,14,19,25,33,21,29,11,16)

Conjugacy classes & character table

9956 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 459);