C₁₇×#11990 #16869734

order	4352
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₇×#11990

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1551
commutator	C₄²	C₂²×C₄×C₁₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₇
prime radical	C₂³×C₁₇	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₁₇×#405 ◁ G

268 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 81 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,28,23,27,41,53,24,30,42,56,44,29,57,55)(20,31,33,59,34,48,63,73,35,47,64,54,43,25,40,46)(21,36,38,65,39,26,49,50,37,52,66,78,67,51,75,77)(32,45,61,71,62,58,79,80,60,72,74,81,69,70,68,76);(18,20,24,35)(19,25,30,48)(21,32,37,60)(22,40,42,63)(23,43,44,34)(26,45,51,72)(27,47,29,31)(28,54,56,59)(33,41,64,57)(36,58,52,70)(38,68,66,79)(39,69,67,62)(46,53,73,55)(49,74,75,61)(50,76,77,80)(65,71,78,81);(18,21)(19,26,30,51)(20,32)(22,38)(23,39)(24,37)(25,45,48,72)(27,52,29,36)(28,50,56,77)(31,58,47,70)(33,61)(34,62)(35,60)(40,68)(41,49)(42,66)(43,69)(44,67)(46,71,73,81)(53,78,55,65)(54,76,59,80)(57,75)(63,79)(64,74)

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, 4978);