C₁₃₇×#9512 #46858898

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₃₇×#9512

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#386
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

558 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 118 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:

(1,2,6,11,5,3,8,14)(4,9,16,25,15,12,20,27)(7,10,21,22,18,13,23,19)(17,26,30,31,28,24,32,29)(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);(1,3,6,19,5,2,8,22)(4,12,20,31,15,9,16,29)(7,13,21,14,18,10,23,11)(17,24,32,25,28,26,30,27);(1,4,7,17)(2,9,10,26)(3,12,13,24)(5,15,18,28)(6,20,21,32)(8,16,23,30)(11,27,22,29)(14,25,19,31)

Conjugacy classes & character table

5069 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 2500);