C₇₉×#12107 #34123796

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#12107

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₇₉×#3191	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₂²⋊C₈×C₇₉ ◁ G

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

normal subgroups only

This group has 428 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) = 5056

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

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

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, 5095);