C₁₇×#7229 #16864973

order	4352
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7229

Subgroups

	subgroup	G/H
center	C₄×C₁₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₇
Frattini, cosocle	C₂×#131	C₂²×C₁₇
socle	C₂×C₁₇	#1289
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₈×C₁₇ ◁ G

244 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 81
transitive degree	μ_t(G) = 1088

A minimal faithful permutation representation — 2 generators on 81 points:

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

Conjugacy classes & character table

680 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 217);