C₇₉×#12068 #34123757

order	20224
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₉×#12068

Subgroups

	subgroup	G/H
center	C₄×C₇₉	#442
commutator	C₈	C₂²×C₈×C₇₉
Frattini, cosocle	C₄×C₈	C₂³×C₇₉
socle	C₂×C₇₉	#1580
prime radical	#508×C₇₉	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₉ ◁ C₂×C₈×C₇₉ ◁ #346×C₇₉ ◁ G

204 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 111
transitive degree	μ_t(G) = 2528

A minimal faithful permutation representation — 3 generators on 111 points:

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

Conjugacy classes & character table

3634 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20224, 5056);