C₇₉×#9429 #34121118

order	20224
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₇₉×#9429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	#532×C₇₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

508 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 5056

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

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

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, 2417);