C₇₉×#12106 #34123795

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

Aliases

direct product

C₇₉×#12106

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₂⁴×C₇₉	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂×C₂²⋊C₈×C₇₉ ◁ G

424 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,10,14,18)(11,19,22,38)(12,23,27,37)(13,28,25,17)(15,30,32,20)(16,33,35,26)(21,39,29,34)(24,36,40,31)(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)(9,11,14,22)(10,15,18,32)(12,24,27,40)(13,29,25,21)(16,34,35,39)(17,36,28,31)(19,37,38,23)(20,33,30,26);(1,3)(2,5)(4,7)(6,8)

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, 5094);