C₇₉×#13723 #34125412

order	20224
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#13723

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₉
prime radical	C₇₉×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₄²×C₇₉ ◁ C₄×C₈×C₇₉ ◁ C₄×C₁₆×C₇₉ ◁ G

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 119
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 119 points:

(1,2)(3,5)(4,6)(7,8)(10,17)(11,22)(12,13)(14,26)(15,31)(18,28)(19,33)(20,36)(21,25)(23,24)(27,35)(29,30)(32,40)(34,37)(38,39)(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);(1,3,4,7)(2,5,6,8)(9,10,18,24,14,20,27,34,31,37,30,36,40,23,12,17,16,22,13,19,32,38,29,25,15,21,35,39,26,33,28,11);(2,6)(5,8)(9,11,12,23,26,39,30,37,15,25,27,20,32,19,18,10,16,17,28,33,40,36,35,21,31,34,29,38,14,24,13,22)

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 6711);