C₁₃₇×#10385 #46859771

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

Aliases

direct product

C₁₃₇×#10385

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

396 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

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, 3373);