C₇₉×#8529 #34120218

order	20224
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#8529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#379
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₇₉×#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

648 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 5056

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

(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(17,18)(19,27,30,40)(20,31,33,24)(21,25)(22,26)(23,29,38,34)(28,43)(32,41,35,39)(36,42)(37,44)(45,48)(46,47)(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);(1,2,6,9)(3,8,11,16)(4,12,5,7)(10,15,13,14)(17,19)(18,23)(20,28)(21,34)(22,30)(24,37)(25,40)(26,38)(27,42)(29,36)(31,47)(32,46)(33,45)(35,44)(39,43)(41,48);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,20)(18,24)(19,28)(21,35)(22,33)(23,37)(25,41)(26,31)(27,43)(29,46)(30,45)(32,36)(34,44)(38,47)(39,42)(40,48)

Conjugacy classes & character table

3160 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 1517);