C₁₇×#9005 #16866749

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

Aliases

direct product

C₁₇×#9005

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

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

normal subgroups only

This group has 416 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)(18,19,22,27,23,25,21,26)(20,28,31,24,30,33,29,32)(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)(19,24)(21,31)(22,29)(23,30)(25,32)(26,33)(27,28)(34,36,38,52,40,48,39,47)(35,41,43,56,44,57,53,45)(37,46,50,63,51,64,54,62)(42,55,58,65,59,61,49,60);(19,25)(21,22)(24,32)(29,31)(34,37)(35,42)(36,46)(38,50)(39,54)(40,51)(41,55)(43,58)(44,59)(45,60)(47,62)(48,64)(49,53)(52,63)(56,65)(57,61)

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, 1993);