C₇₉×#9335 #34121024

order	20224
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇₉×#9335

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	#532×C₇₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

400 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

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, 2323);