C₁₃₁×#12263 #45698147

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

Aliases

direct product

C₁₃₁×#12263

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃₁
prime radical	#392×C₁₃₁	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₂²⋊C₄×C₁₃₁ ◁ G

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

normal subgroups only

This group has 438 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:

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

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, 5251);