C₁₃₁×#7239 #45693123

order	33536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₁×#7239

Subgroups

	subgroup	G/H
center	C₄×C₁₃₁	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₃₁
Frattini, cosocle	#405	C₂²×C₁₃₁
socle	C₂×C₁₃₁	#1290
prime radical	C₂³×C₁₃₁	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₁ ◁ C₂×C₈×C₁₃₁ ◁ G

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

normal subgroups only

This group has 148 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) = 195
transitive degree	μ_t(G) = 8384

A minimal faithful permutation representation — 2 generators on 195 points:

(1,2,5,11,7,12,27,40,8,13,28,41,30,36,6,29)(3,14,16,43,18,45,17,51,19,46,25,44,52,62,53,63)(4,20,21,54,23,39,22,38,24,56,26,55,57,10,35,37)(9,31,32,48,33,59,60,49,34,50,61,64,42,47,58,15)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195);(1,3,6,17,30,52,28,16,8,19,27,53,7,18,5,25)(2,9,11,32,12,33,40,60,13,34,41,61,36,42,29,58)(4,15,22,47,57,64,21,50,24,49,35,59,23,48,26,31)(10,14,38,44,20,45,55,63,39,46,37,43,56,62,54,51)

Conjugacy classes & character table

5240 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33536, 227);