C₁₇×#7208 #16864952

order	4352
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#7208

Subgroups

	subgroup	G/H
center	C₈×C₁₇	#99
commutator	C₂×C₄	C₄×C₈×C₁₇
Frattini, cosocle	C₂×#131	C₂²×C₁₇
socle	C₂×C₁₇	#1282
prime radical	C₂×#41×C₁₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂×C₈×C₁₇ ◁ C₂×C₁₇×#131 ◁ G

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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)(18,19,22,28,24,30,43,53,25,29,23,44,42,54,45,70)(20,31,33,55,34,58,62,78,35,57,40,56,63,77,67,79)(21,36,37,51,38,64,65,76,39,27,41,50,66,52,68,75)(26,46,47,60,49,72,73,61,48,59,69,71,74,80,81,32);(18,20,23,33,24,34,45,62,25,35,22,40,42,63,43,67)(19,26,28,47,30,49,53,73,29,48,44,69,54,74,70,81)(21,32,37,59,38,60,65,80,39,61,41,46,66,71,68,72)(27,31,51,56,52,58,76,79,36,57,50,55,64,77,75,78)

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, 196);