C₇₉×#9424 #34121113

order	20224
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#9424

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂×#127×C₇₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

436 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 111
transitive degree	μ_t(G) = 5056

A minimal faithful permutation representation — 3 generators on 111 points:

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

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, 2412);