C₇₉×#7477 #34119166

order	20224
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#357
commutator	C₁₆	C₄²×C₇₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇₉
prime radical	C₂³×C₇₉	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×C₈×C₇₉ ◁ C₂²×C₁₆×C₇₉ ◁ G

178 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 115
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 2 generators on 115 points:

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

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, 465);