C₁₇×#9008 #16866752

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

Aliases

direct product

C₁₇×#9008

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₇	#120
commutator	C₄²	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
socle	C₂²×C₁₇	#386
prime radical	C₁₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

398 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 65
transitive degree	μ_t(G) = 2176

A minimal faithful permutation representation — 3 generators on 65 points:

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

Conjugacy classes & character table

680 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 1996);