C₁₇×#12365 #16870109

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12364
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#12365

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₇
prime radical	C₁₇×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₄×C₈×C₁₇ ◁ G

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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)(20,29)(21,32,34,49)(22,27)(23,37)(24,39)(25,31)(26,30,43,48)(28,36)(33,41,40,46)(35,44)(38,45,47,42);(18,20,22,31)(19,25,27,29)(21,30,34,48)(23,24,35,36)(26,41,43,46)(28,44,39,37)(32,42,49,45)(33,38,40,47);(18,21)(19,26)(20,30)(22,34)(23,38)(24,40)(25,41)(27,43)(28,45)(29,46)(31,48)(32,44)(33,36)(35,47)(37,49)(39,42)

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, 5353);