C₁₃₇×#9983 #46859369

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₇×#9983

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	#2815×C₁₃₇	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

602 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 169
transitive degree	μ_t(G) = 4384

A minimal faithful permutation representation — 3 generators on 169 points:

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

Conjugacy classes & character table

3836 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 2971);