C₁₇×#13718 #16871462

order	4352
nilpotency class	5
isoclinism	isoclinic class of #13717
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#13718

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#2233
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₁₇
prime radical	C₂³×C₁₇	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ C₂×C₁₇×M₃₂ ◁ G

160 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 81 points:

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

Conjugacy classes & character table

527 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 6706);