C₇₁×#8546 #31829418

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇₁ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₇₁×#8546

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#379
commutator	C₂⁴	C₂²×C₄×C₇₁
Frattini, cosocle	C₂⁵	C₂³×C₇₁
prime radical	C₂²×#127×C₇₁	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂⁵×C₇₁ ◁ G

1846 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1846 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 87
transitive degree	μ_t(G) = 2272

A minimal faithful permutation representation — 3 generators on 87 points:

(1,2)(3,7,8,5)(4,6)(9,10)(11,15,16,13)(12,14)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(1,3,4,8)(2,5,6,7);(9,11,12,16)(10,13,14,15)

Conjugacy classes & character table

2840 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 1534);