C₁₃₁×#11490 #45697374

order	33536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₃₁×#11490

Subgroups

	subgroup	G/H
center	C₂×C₈×C₁₃₁	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂²×C₁₃₁	C₂×C₂²⋊C₈
prime radical	#532×C₁₃₁	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

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

normal subgroups only

This group has 364 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) = 171
transitive degree	μ_t(G) = 16768

A minimal faithful permutation representation — 3 generators on 171 points:

(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,28,36,15,21,27,37,13,29,12,18)(11,22,25,32,24,17,31,33,26,35,39,40,30,34,23,38)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171);(2,5)(3,7)(9,11)(10,17)(12,23)(13,30)(14,25)(15,26)(16,24)(18,32)(19,35)(20,33)(21,34)(22,29)(27,39)(28,31)(36,40)(37,38);(9,12,13,27,15,28,16,14)(10,18,29,37,21,36,19,20)(11,23,30,39,26,31,24,25)(17,32,22,38,34,40,35,33)

Conjugacy classes & character table

9956 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 4478);