C₇₉×#33728 #34145417

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₇₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₇₉×#33728

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₉
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂³×C₇₉	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

482 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 270 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) = 119
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 4 generators on 119 points:

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

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, 26716);