C₁₇×#7228 #16864972

order	4352
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₇×#7228

Subgroups

	subgroup	G/H
center	C₄×C₁₇	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₇
Frattini, cosocle	C₂×#131	C₂²×C₁₇
socle	C₂×C₁₇	#1289
prime radical	C₂×#41×C₁₇	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₂×C₈×C₁₇ ◁ G

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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)(20,30)(21,35,37,61)(22,40)(23,28)(24,29)(25,31,43,55)(26,39,46,63)(27,48)(32,58)(33,52)(34,53)(36,50)(38,65,64,47)(41,49)(42,68)(44,71,69,79)(45,57,70,76)(51,72)(54,75)(56,78,77,67)(59,74)(60,80)(62,66,81,73);(18,20,22,32,36,54,48,72,24,34,42,60,23,33,41,59)(19,25,27,44,28,45,40,67,29,43,49,69,50,70,68,78)(21,31,38,56,63,76,73,71,37,55,64,77,39,57,66,79)(26,30,47,51,35,52,62,58,46,53,65,74,61,75,81,80)

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, 216);