C₁₇×#13565 #16871309

order	4352
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₇×#13565

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₇
Frattini, cosocle	#124	C₂³×C₁₇
prime radical	C₂×C₁₇×H(5,𝔽₂)	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂³×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ C₂×C₁₇×#124 ◁ G

496 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 49
transitive degree	μ_t(G) = 544

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,21,26,23,28,35,44)(20,29,31,42,30,45,46,49)(22,27,38,43,37,25,34,41)(24,33,40,36,39,47,48,32);(18,20)(19,24)(21,33)(22,36)(23,30)(25,29)(26,42)(27,45)(28,39)(31,38)(32,37)(34,46)(35,47)(40,43)(41,48)(44,49);(20,30)(24,39)(29,45)(31,46)(32,36)(33,47)(40,48)(42,49)

Conjugacy classes & character table

476 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 6553);