C₁₃₁×#7265 #45693149

order	33536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₁×#7265

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	#340
commutator	C₂×C₈	C₄²×C₁₃₁
Frattini, cosocle	C₂³×C₈	C₂²×C₁₃₁
prime radical	C₂⁴×C₁₃₁	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

314 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 8384

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

(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,20,14,19)(11,21,23,38,25,39,24,35)(12,26,28,37,30,36,29,17)(16,31,32,22,34,27,33,40)(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);(2,5,7,6)(9,11,13,24,15,25,14,23)(10,16,18,33,20,34,19,32)(12,27,28,22,30,31,29,40)(17,35,26,39,37,38,36,21)

Conjugacy classes & character table

6026 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 253);