C₁₃₁×#12179 #45698063

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

Aliases

direct product

C₁₃₁×#12179

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₁	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃₁
prime radical	#534×C₁₃₁	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₂²⋊C₄×C₁₃₁ ◁ G

412 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 155
transitive degree	μ_t(G) = 8384

A minimal faithful permutation representation — 3 generators on 155 points:

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

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, 5167);