C₁₇×#12118 #16869862

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#12118

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1595
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₇
prime radical	C₁₇×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂×M₁₆×C₁₇ ◁ G

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

normal subgroups only

This group has 392 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)(20,32)(21,37)(22,41)(23,30)(24,29)(25,31)(26,34)(27,39)(28,44)(33,61)(35,69)(36,60)(38,72)(40,57)(42,50)(43,53)(45,58)(46,54)(47,56)(48,55)(49,76)(51,67)(52,68)(59,63)(62,70)(64,81)(65,79)(66,77)(71,75)(73,78)(74,80);(18,20,25,36)(19,26,31,52)(21,33,40,65)(22,42,45,67)(23,35,24,48)(27,49,54,78)(28,55,56,69)(29,51,30,50)(32,58,60,41)(34,47,68,44)(37,70,57,77)(38,73,43,76)(39,63,46,64)(53,79,72,61)(59,80,81,71)(62,74,66,75);(18,21)(19,27)(20,33)(22,43)(23,46)(24,39)(25,40)(26,49)(28,53)(29,57)(30,37)(31,54)(32,59)(34,66)(35,64)(36,65)(38,45)(41,71)(42,76)(44,74)(47,75)(48,63)(50,70)(51,77)(52,78)(55,79)(56,72)(58,80)(60,81)(61,69)(62,68)(67,73)

Conjugacy classes & character table

425 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5106);