C₁₇×#13199 #16870943

order	4352
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#13199

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#124
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

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

normal subgroups only

This group has 406 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)(18,19,23,29,24,30,45,59,25,31,46,60,22,28,33,48)(20,32,37,61,35,50,62,58,36,56,44,72,42,26,47,51)(21,27,40,54,39,53,68,77,41,55,71,80,43,57,63,74)(34,64,67,79,38,49,69,76,66,52,73,78,65,75,70,81);(18,20,33,62,22,42,46,37,25,36,45,47,24,35,23,44)(19,26,48,61,28,56,60,51,31,50,59,72,30,32,29,58)(21,38,68,67,43,34,40,70,41,65,63,73,39,66,71,69)(27,52,77,76,57,49,54,79,55,64,74,81,53,75,80,78);(18,21)(19,27)(20,34,36,66)(22,43)(23,40)(24,39)(25,41)(26,49,50,75)(28,57)(29,54)(30,53)(31,55)(32,52,56,64)(33,63)(35,38,42,65)(37,67,44,73)(45,68)(46,71)(47,70,62,69)(48,74)(51,76,58,81)(59,77)(60,80)(61,78,72,79)

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, 6187);