C₇₉×#8791 #34120480

order	20224
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#8791

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#382
commutator	C₄²	C₂²×C₄×C₇₉
Frattini, cosocle	C₂×C₄²	C₂³×C₇₉
prime radical	C₂³×C₇₉	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Conjugacy classes & character table

4108 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 1779);