C₁₃₇×#8140 #46857526

order	35072
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₃₇ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₁₃₇×#8140

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle	C₂²×C₁₃₇	#376
prime radical	C₂³×C₁₃₇	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₇ ◁ G

330 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Conjugacy classes & character table

8768 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35072, 1128);