C₁₃₇×#7085 #46856471

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

Aliases

direct product

C₁₃₇×#7085

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#327
commutator	C₂²×C₈	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₁₃₇
prime radical	C₂⁴×C₁₃₇	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×C₂²⋊C₈×C₁₃₇ ◁ G

196 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 169 points:

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

Conjugacy classes & character table

4658 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 73);