C₇₉×#7827 #34119516

order	20224
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#7827

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 358 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) = 119
transitive degree	μ_t(G) = 20224

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

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

Conjugacy classes & character table

2923 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 815);