C₇₁×#7776 #31828648

order	18176
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇₁×#7776

Subgroups

	subgroup	G/H
center, socle	C₂³×C₇₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂²×D₄×C₇₁	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

446 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 95
transitive degree	μ_t(G) = 9088

A minimal faithful permutation representation — 3 generators on 95 points:

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

Conjugacy classes & character table

2627 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 764);