C₁₃₉×#7517 #48002113

order	35584
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₃₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₉×#7517

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₃₉	#1402
commutator	#127	C₂×C₄×C₁₃₉
Frattini, cosocle	#492	C₂²×C₁₃₉
prime radical	D₄²×C₁₃₉	C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₉ ◁ C₂²×C₁₃₉ ◁ C₂×C₄×C₁₃₉ ◁ C₄²×C₁₃₉ ◁ #492×C₁₃₉ ◁ G

196 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 155
transitive degree	μ_t(G) = 2224

A minimal faithful permutation representation — 2 generators on 155 points:

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

Conjugacy classes & character table

3058 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35584, 505);