C₁₇×#13078 #16870822

order	4352
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#13078

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1893
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂×C₁₇×#131 ◁ G

454 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 146 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)(18,19,23,30,25,29,24,48)(20,31,35,61,34,58,46,62)(21,36,38,57,39,27,47,56)(22,40,43,50,45,72,44,28)(26,49,52,33,54,68,53,41)(32,60,65,80,64,77,66,81)(37,70,71,59,42,73,74,55)(51,76,78,67,69,75,79,63);(18,20,22,41,24,35,44,49,25,34,45,33,23,46,43,68)(19,26,50,31,30,53,72,62,29,54,28,58,48,52,40,61)(21,32,42,63,38,66,74,75,39,64,37,67,47,65,71,76)(27,51,55,60,57,78,73,80,36,69,59,77,56,79,70,81);(18,21)(19,27)(20,32)(22,42)(23,47)(24,38)(25,39)(26,51)(28,59)(29,36)(30,57)(31,60)(33,67)(34,64)(35,66)(37,45)(40,70)(41,63)(43,71)(44,74)(46,65)(48,56)(49,75)(50,55)(52,79)(53,78)(54,69)(58,77)(61,81)(62,80)(68,76)(72,73)

Conjugacy classes & character table

782 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 6066);