C₇₉×#7429 #34119118

order	20224
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇₉×#7429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#354
commutator	C₂×C₄²	C₂×C₄×C₇₉
Frattini, cosocle	#391	C₂²×C₇₉
prime radical	C₂³×C₇₉	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂³×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #391×C₇₉ ◁ G

196 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 119 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,19,26,34)(11,20,22,31,24,35,39,15)(13,27,28,32,29,18,25,16)(21,36,37,33,38,30,23,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5,6,7)(3,8)(9,11,13,23,14,24,29,37)(10,15,18,30,19,31,27,36)(12,21,25,22,26,38,28,39)(16,20,17,33,32,35,34,40)

Conjugacy classes & character table

2212 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 417);