C₇₉×#18732 #34130421

order	20224
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₇₉×#18732

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₉
prime radical	C₂³×C₇₉	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

808 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 127 points:

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

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, 11720);