C₁₃₉×#7067 #48001663

order	35584
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₃₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₃₉×#7067

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₃₉	#99
commutator	C₂²×C₄	C₂×C₈×C₁₃₉
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₁₃₉
socle	C₂²×C₁₃₉	#324
prime radical	C₂⁴×C₁₃₉	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×C₂²⋊C₈×C₁₃₉ ◁ G

210 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 163
transitive degree	μ_t(G) = 8896

A minimal faithful permutation representation — 2 generators on 163 points:

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

Conjugacy classes & character table

6394 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35584, 55);