C₁₃₇×#9509 #46858895

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₃₇×#9509

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#386
commutator	C₂⁴	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

524 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 8768

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

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

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, 2497);