C₇₁×#12066 #31832938

order	18176
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#12066

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₇₁	#442
commutator	C₈	C₂²×C₈×C₇₁
Frattini, cosocle	C₄×C₈	C₂³×C₇₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ #322×C₇₁ ◁ G

214 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 18176

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

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

Conjugacy classes & character table

3266 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 5054);