C₇₉×#12021 #34123710

order	20224
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₇₉×#12021

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
prime radical	C₂³×C₇₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂×C₄²×C₇₉ ◁ C₂×C₄×C₈×C₇₉ ◁ G

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

normal subgroups only

This group has 216 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) = 119
transitive degree	μ_t(G) = 10112

A minimal faithful permutation representation — 3 generators on 119 points:

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

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, 5009);