C₁₃₇×#20321 #46869707

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃₇×#20321

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

542 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 35072

A minimal faithful permutation representation — 4 generators on 169 points:

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

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, 13309);