C₁₃₁×#12724 #45698608

order	33536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃₁×#12724

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃₁	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₃₁
Frattini, cosocle	C₂×#41	C₂³×C₁₃₁
prime radical	C₂×#41×C₁₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₂×C₄×C₁₃₁ ◁ C₄×#41×C₁₃₁ ◁ G

670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 163
transitive degree	μ_t(G) = 4192

A minimal faithful permutation representation — 3 generators on 163 points:

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

Conjugacy classes & character table

4454 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 5712);