C₁₇×#7365 #16865109

order	4352
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7365

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#99
commutator	C₂³	C₂×C₁₆×C₁₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₇
socle	C₂×C₁₇	#1322
prime radical	C₂³×C₁₇	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂×C₈×C₁₇ ◁ C₂×C₁₇×M₃₂ ◁ G

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 81 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,27,22,28,35,48,23,29,36,49,37,50,59,71,24,26,34,44,38,45,57,66,39,46,58,67,60,68,76,79)(20,25,31,40,32,42,54,61,33,43,55,62,56,65,75,77,30,41,51,47,52,63,72,69,53,64,73,70,74,78,81,80);(18,20)(19,25,26,41)(21,31)(22,32)(23,33)(24,30)(27,47,44,40)(28,42,45,63)(29,43,46,64)(34,51)(35,54)(36,55)(37,56)(38,52)(39,53)(48,69,66,61)(49,70,67,62)(50,65,68,78)(57,72)(58,73)(59,75)(60,74)(71,80,79,77)(76,81)

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, 353);