C₁₃₁×#29657 #45715541

order	33536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₃₁×#29657

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₁
prime radical	C₂⁵×C₁₃₁	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂⁵×C₁₃₁ ◁ G

1148 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 242 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 — 4 generators on 163 points:

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

Conjugacy classes & character table

4847 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 22645);