C₇₉×#12897 #34124586

order	20224
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12897

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇₉	#117
commutator	C₂×C₄	C₂²×C₈×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
socle	C₂²×C₇₉	#432
prime radical	C₂³×C₇₉	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₉ ◁ C₂×#97×C₇₉ ◁ G

278 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 5056

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

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

Conjugacy classes & character table

5056 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5885);