C₁₃₇×#7365 #46856751

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₇×#7365

Subgroups

	subgroup	G/H
center	C₈×C₁₃₇	#99
commutator	C₂³	C₂×C₁₆×C₁₃₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₃₇
socle	C₂×C₁₃₇	#1322
prime radical	C₂³×C₁₃₇	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ C₂×M₃₂×C₁₃₇ ◁ G

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 201
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 2 generators on 201 points:

(1,2,4,10,5,11,18,31,6,12,19,32,20,33,42,54,7,9,17,27,21,28,40,49,22,29,41,50,43,51,59,62)(3,8,14,23,15,25,37,44,16,26,38,45,39,48,58,60,13,24,34,30,35,46,55,52,36,47,56,53,57,61,64,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201);(1,3)(2,8,9,24)(4,14)(5,15)(6,16)(7,13)(10,30,27,23)(11,25,28,46)(12,26,29,47)(17,34)(18,37)(19,38)(20,39)(21,35)(22,36)(31,52,49,44)(32,53,50,45)(33,48,51,61)(40,55)(41,56)(42,58)(43,57)(54,63,62,60)(59,64)

Conjugacy classes & character table

7124 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 353);