C₇₁×#7777 #31828649

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

Aliases

direct product

C₇₁×#7777

Subgroups

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

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

normal subgroups only

This group has 392 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) = 103
transitive degree	μ_t(G) = 18176

A minimal faithful permutation representation — 3 generators on 103 points:

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

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, 765);