C₁₃₇×#7097 #46856483

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₇×#7097

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	#331
commutator	C₄×C₈	C₂×C₄×C₁₃₇
Frattini, cosocle	C₈²	C₂²×C₁₃₇
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₈²×C₁₃₇ ◁ G

142 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 169 points:

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

Conjugacy classes & character table

5891 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 85);