C₇₉×#13497 #34125186

order	20224
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#13497

Subgroups

	subgroup	G/H
center, socle	C₂×C₇₉	#2075
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₂×#41×C₇₉	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×#131×C₇₉ ◁ G

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

normal subgroups only

This group has 274 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

2686 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6485);