C₇₉×#9422 #34121111

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×#9422

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂³×C₇₉	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

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

normal subgroups only

This group has 384 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) = 127
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 127 points:

(1,2,6,10,7,9,5,3)(4,12,14,16,15,8,13,11)(17,18,22,27,23,26,21,34)(19,24,30,39,29,40,36,38)(20,31,32,25,33,42,35,41)(28,43,46,47,44,48,45,37)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,7,10)(2,6,9,5)(4,11,15,16)(8,13,12,14)(17,19,21,30,23,29,22,36)(18,24,27,38,26,40,34,39)(20,28,35,46,33,44,32,45)(25,37,42,48,41,47,31,43);(1,4,7,15)(2,8,9,12)(3,11,10,16)(5,14,6,13)(17,20,23,33)(18,25,26,41)(19,28,29,44)(21,35,22,32)(24,37,40,47)(27,42,34,31)(30,46,36,45)(38,48,39,43)

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, 2410);