C₁₇×#12466 #16870210

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#12466

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₇
prime radical	C₁₇×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₄×C₈×C₁₇ ◁ G

186 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 70 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)(18,19,26,33)(20,34,41,66)(21,42,46,58)(22,30,47,52)(23,29,35,61)(24,32,25,31)(27,51,57,37)(28,43,60,44)(36,68,71,80)(38,55,48,56)(39,54,40,62)(45,75,50,63)(49,64,67,59)(53,74,78,70)(65,72,77,69)(73,79,76,81);(18,20,35,38,24,39,22,37,26,41,23,48,25,40,47,51)(19,27,52,54,31,55,29,34,33,57,30,62,32,56,61,66)(21,36,67,73,45,70,44,72,46,71,49,76,50,74,43,69)(28,53,75,79,59,68,42,65,60,78,63,81,64,80,58,77);(18,21)(19,28)(20,36)(22,44)(23,49)(24,45)(25,50)(26,46)(27,53)(29,42)(30,63)(31,59)(32,64)(33,60)(34,65)(35,67)(37,72)(38,73)(39,70)(40,74)(41,71)(43,47)(48,76)(51,69)(52,75)(54,79)(55,68)(56,80)(57,78)(58,61)(62,81)(66,77)

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, 5454);