C₁₇×#7378 #16865122

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

Aliases

direct product

C₁₇×#7378

Subgroups

	subgroup	G/H
center	C₄×C₁₇	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₁₇
Frattini, cosocle	#408	C₂²×C₁₇
socle	C₂×C₁₇	#1326
prime radical	C₂×#41×C₁₇	C₂×C₄
LCS	G ▷ C₂×Q₈ ▷ 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)(18,19,22,28,24,29,43,59,25,30,44,60,23,45,47,58)(20,31,33,61,35,63,41,74,36,53,68,77,34,62,70,80)(21,37,38,71,39,27,52,54,40,55,42,75,46,56,73,81)(26,48,49,67,50,76,78,69,51,72,79,32,64,65,57,66);(18,20,23,34,25,36,24,35)(19,26,29,50,30,51,45,64)(21,32,39,66,40,67,46,69)(22,41,43,68,44,70,47,33)(27,53,37,63,56,31,55,62)(28,57,58,79,60,78,59,49)(38,72,73,76,42,48,52,65)(54,80,75,61,81,74,71,77)

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, 366);