C₁₃₇×#11543 #46860929

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₇×#11543

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×#99
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₁₃₇
prime radical	C₂⁵×C₁₃₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂⁴×C₁₃₇ ◁ C₂²×C₂²⋊C₄×C₁₃₇ ◁ G

492 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

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, 4531);