C₁₇×#7425 #16865169

order	4352
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7425

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1347
commutator	C₂×C₁₆	C₂×C₄×C₁₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₇
prime radical	C₂³×C₁₇	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ C₂×C₁₇×M₃₂ ◁ G

136 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 46 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)(18,19,22,29,26,31,25,52)(20,32,34,66,38,67,37,73)(21,39,42,74,45,65,46,75)(23,48,47,64,24,50,41,30)(27,53,54,69,57,70,56,33)(28,58,60,43,62,77,61,44)(35,71,49,80,36,72,51,59)(40,63,76,79,68,81,78,55);(18,20,26,38)(19,27,31,57)(21,40,45,68)(22,37,25,34)(23,49,24,51)(28,59,62,80)(29,54,52,56)(30,63,64,81)(32,65,67,39)(33,43,69,44)(35,41,36,47)(42,78,46,76)(48,79,50,55)(53,58,70,77)(60,71,61,72)(66,75,73,74)

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, 413);