C₇₉×#7413 #34119102

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×#7413

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

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

normal subgroups only

This group has 178 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)(3,5)(4,7)(6,8)(9,10,12,17,14,19,27,37)(11,20,22,15,24,32,40,34)(13,28,30,35,31,18,26,16)(21,38,23,36,25,33,29,39)(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);(2,5)(3,8)(6,7)(9,11)(10,15,19,34)(12,25)(13,29)(14,24)(16,20,35,32)(17,36,37,39)(18,38,28,33)(21,27)(22,30)(23,31)(26,40)

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, 401);