C₁₃₇×#13444 #46862830

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

Aliases

direct product

C₁₃₇×#13444

Subgroups

	subgroup	G/H
center	C₄×C₁₃₇	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₇
socle	C₂×C₁₃₇	#2054
prime radical	#3414×C₁₃₇	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×C₄×C₈×C₁₃₇ ◁ G

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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)(3,4)(5,18)(6,26)(7,28)(8,13)(9,14)(10,20)(11,25)(12,24)(15,30)(16,32)(17,22)(19,23)(21,29)(27,31)(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);(1,3)(2,9)(4,18)(5,23)(6,10)(7,29)(8,17)(11,21)(12,19)(13,30)(14,25)(15,27)(16,24)(20,28)(22,26)(31,32);(1,4,5,19,24,32,27,30,8,22,6,20,7,21,25,9)(2,10,11,17,28,15,26,16,13,23,31,3,12,14,18,29)

Conjugacy classes & character table

5480 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 6432);