C₁₃₁×#12268 #45698152

order	33536
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₁×#12268

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃₁
prime radical	C₂⁵×C₁₃₁	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₂²⋊C₄×C₁₃₁ ◁ G

458 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

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, 5256);