C₇₉×#7218 #34118907

order	20224
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₇₉×#7218

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	#340
commutator	C₂×C₈	C₄²×C₇₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₉
prime radical	C₂³×C₇₉	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

202 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 2 generators on 127 points:

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

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, 206);