C₁₇×#13198 #16870942

order	4352
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#13198

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1955
prime radical	C₁₇×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ G

410 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 138 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)(18,19,22,27,23,28,38,49,24,29,39,50,40,51,64,75)(20,25,32,43,33,44,58,70,31,42,56,68,57,69,79,81)(21,26,36,47,37,48,53,66,35,46,60,72,63,74,78,76)(30,52,55,77,34,59,62,80,54,41,65,67,61,45,71,73);(18,20,22,32,23,33,38,58,24,31,39,56,40,57,64,79)(19,25,27,43,28,44,49,70,29,42,50,68,51,69,75,81)(21,34,60,71,37,54,78,55,35,61,36,62,63,30,53,65)(26,45,72,80,48,52,76,67,46,59,47,73,74,41,66,77);(18,21)(19,26)(20,30,31,54)(22,36)(23,37)(24,35)(25,41,42,52)(27,47)(28,48)(29,46)(32,55,56,65)(33,34,57,61)(38,53)(39,60)(40,63)(43,67,68,77)(44,45,69,59)(49,66)(50,72)(51,74)(58,62,79,71)(64,78)(70,73,81,80)(75,76)

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 6186);