C₁₇×#7377 #16865121

order	4352
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7377

Subgroups

	subgroup	G/H
center	C₄×C₁₇	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂×C₈×C₁₇
Frattini, cosocle	#408	C₂²×C₁₇
socle	C₂×C₁₇	#1326
prime radical	C₂×#41×C₁₇	C₂×C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₁₇×#408 ◁ G

170 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 46 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)(18,19,22,28,24,29,44,59,25,30,45,60,23,46,48,58)(20,31,33,61,35,63,67,81,36,64,68,77,34,62,42,74)(21,37,39,70,40,27,53,54,41,55,43,75,47,56,73,80)(26,49,50,32,51,65,57,66,52,76,79,38,71,72,78,69);(18,20,23,34,25,36,24,35)(19,26,29,51,30,52,46,71)(21,38,40,69,41,32,47,66)(22,42,44,33,45,67,48,68)(27,31,37,62,56,64,55,63)(28,57,58,50,60,78,59,79)(39,49,73,72,43,76,53,65)(54,74,75,61,80,81,70,77)

Conjugacy classes & character table

578 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 365);