C₁₃₇×#7422 #46856808

order	35072
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₃₇×#7422

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃₇	#1347
commutator	C₂×C₁₆	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₃₇
prime radical	#2218×C₁₃₇	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂²×C₈×C₁₃₇ ◁ C₂×M₃₂×C₁₃₇ ◁ G

150 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 169 points:

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

Conjugacy classes & character table

4247 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 410);