C₁₃₇×#7111 #46856497

order	35072
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₃₇×#7111

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#327
commutator	C₂×C₈	C₂×C₈×C₁₃₇
Frattini, cosocle	C₂×#97	C₂²×C₁₃₇
prime radical	C₂³×C₁₃₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ C₂×#97×C₁₃₇ ◁ G

234 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 2 generators on 177 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,19,15,22,31,18,16,20,30,37,14,21,12,29)(11,23,25,34,27,39,24,35,28,38,40,17,26,33,32,36)(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);(2,5)(3,8)(6,7)(9,11,16,28)(10,17,20,34)(12,25,31,40)(13,32,30,24)(14,27,15,26)(18,33,29,39)(19,23,37,38)(21,36,22,35)

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, 99);