C₁₃₇×#9982 #46859368

order	35072
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₃₇×#9982

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₇
prime radical	C₂²×#41×C₁₃₇	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂³×C₄×C₁₃₇ ◁ G

722 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 177
transitive degree	μ_t(G) = 8768

A minimal faithful permutation representation — 3 generators on 177 points:

(1,2)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,24,32)(12,17,28,35)(13,19,14,18)(16,25,31,23)(22,37,38,33)(26,40,39,30)(27,36,29,34)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177);(5,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,32)(19,21)(20,31)(22,27)(28,39)(29,38)(30,34)(35,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,28)(10,17,20,35)(11,22,24,38)(13,27,14,29)(16,30,31,40)(18,34,19,36)(21,33,32,37)(23,26,25,39)

Conjugacy classes & character table

3836 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 2970);