C₁₇×#7387 #16865131

order	4352
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#7387

Subgroups

	subgroup	G/H
center	C₈×C₁₇	D₁₆
commutator	C₁₆	C₂×C₈×C₁₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₇
socle	C₂×C₁₇	#1331
prime radical	C₁₇×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₇ ◁ C₂×C₈×C₁₇ ◁ C₄×C₈×C₁₇ ◁ C₄×C₁₆×C₁₇ ◁ G

120 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 54 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)(18,19,22,28,24,29,44,57,25,31,45,58,48,30,23,46)(20,32,34,60,35,61,66,80,37,62,67,81,36,68,47,53)(21,38,40,54,42,69,75,56,43,71,76,55,77,70,41,27)(26,39,49,63,51,72,78,65,52,74,79,64,59,73,50,33);(18,20)(19,26)(21,39)(22,34)(23,47)(24,35)(25,37)(27,53)(28,49)(29,51)(30,59)(31,52)(32,38)(33,41)(36,48)(40,63)(42,72)(43,74)(44,66)(45,67)(46,50)(54,60)(55,81)(56,80)(57,78)(58,79)(61,69)(62,71)(64,76)(65,75)(68,70)(73,77)

Conjugacy classes & character table

1292 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 375);