C₁₃₇×#8750 #46858136

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

Aliases

direct product

C₁₃₇×#8750

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#381
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂⁵	C₂³×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂⁵×C₁₃₇ ◁ G

538 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 110 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)(3,11,13,7)(4,15,17,21)(5,9)(6,10)(8,16,22,32)(12,26,28,24)(14,25,30,20)(18,23)(19,27,31,29)(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);(1,3,6,14)(2,7,10,20)(4,12,17,29)(5,13,18,30)(8,19,22,26)(9,11,23,25)(15,31,21,24)(16,27,32,28);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,24)(10,22,23,15)(13,27)(14,29)(18,32)(20,26,25,31)(28,30)

Conjugacy classes & character table

4658 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 1738);