C₇₉×#7480 #34119169

order	20224
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7480

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#357
commutator	C₁₆	C₄²×C₇₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇₉
socle	C₂×C₇₉	#1380
prime radical	C₂³×C₇₉	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

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

normal subgroups only

This group has 166 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) = 143
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 143 points:

(1,2,5,11,8,15,29,14)(3,16,18,48,21,40,52,50)(4,22,25,46,27,45,63,12)(6,30,28,42,32,41,26,10)(7,33,24,13,31,43,62,44)(9,34,35,23,39,60,38,17)(19,54,53,47,56,37,61,64)(20,57,51,49,55,36,59,58)(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);(1,3,6,19,31,55,27,17,29,52,26,61,24,51,25,60,8,21,32,56,7,20,4,23,5,18,28,53,62,59,63,34)(2,9,12,36,44,54,42,16,11,35,22,58,33,47,41,48,15,39,46,57,13,37,10,40,14,38,45,49,43,64,30,50)

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 468);