C₇₉×#13722 #34125411

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

Aliases

direct product

C₇₉×#13722

Subgroups

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

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

normal subgroups only

This group has 184 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)(9,10)(11,24)(12,17)(13,21)(14,20)(15,25)(16,30)(18,23)(19,26)(22,31)(27,32)(28,34)(29,37)(33,39)(35,40)(36,38)(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)(9,11,14,25,15,20,24,10,18,21,36,22,37,39,19,30,17,27,34,40,35,28,32,12,16,26,33,29,31,38,13,23);(2,6)(5,8)(9,12,13,28,31,40,33,27,16,30,32,39,35,22,34,21,17,10,19,20,37,25,36,11,18,23,24,38,15,29,14,26)

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, 6710);