C₇₁×#10302 #31831174

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

Aliases

direct product

C₇₁×#10302

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	#532×C₇₁	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

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

normal subgroups only

This group has 516 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) = 95
transitive degree	μ_t(G) = 4544

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

(1,2)(3,6)(4,7)(5,8)(9,10,13,17)(11,18,21,19)(12,23,14,16)(15,22,24,20)(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);(1,2)(3,7)(4,6)(5,8)(9,11,13,21)(10,15,17,24)(12,22,14,20)(16,19,23,18);(2,6)(3,4)(7,8)(9,10)(11,19)(12,16)(13,17)(14,23)(15,20)(18,21)(22,24)

Conjugacy classes & character table

2414 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18176, 3290);