C₁₃₇×#12414 #46861800

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₇×#12414

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
prime radical	C₂×D₈×C₁₃₇	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ #446×C₁₃₇ ◁ G

286 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 17536

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

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

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, 5402);