C₁₇×#7423 #16865167

order	4352
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7423

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	#1347
commutator	C₂×C₁₆	C₂×C₄×C₁₇
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₇
prime radical	C₂³×C₁₇	D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂×C₄×C₁₇ ◁ C₂²×C₈×C₁₇ ◁ C₂×C₁₇×M₃₂ ◁ G

140 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 46 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)(20,31,33,62)(21,37,40,61)(22,44)(23,47)(24,50)(25,30)(26,32)(27,41,56,42)(28,39)(29,46)(34,67,35,68)(36,69,51,63)(38,58)(43,76,74,70)(45,79,48,55)(49,59)(52,77)(53,71)(54,66)(57,75,80,73)(60,65)(64,81)(72,78);(18,20,25,33)(19,26,30,54)(21,38,40,64)(22,45,23,48)(24,51,46,36)(27,55,56,79)(28,58,59,81)(29,60,50,53)(31,61,62,37)(32,41,66,42)(34,39,35,49)(43,77,74,72)(44,78,47,52)(57,68,80,67)(63,70,69,76)(65,75,71,73)

Conjugacy classes & character table

527 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 411);