C₁₇×#12027 #16869771

order	4352
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₇×#12027

Subgroups

	subgroup	G/H
center	C₄×C₁₇	#427
commutator	C₂×C₈	C₂²×C₄×C₁₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₇
socle	C₂×C₁₇	#1568
prime radical	C₂³×C₁₇	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₇ ◁ C₁₇×#97 ◁ C₁₇×#405 ◁ G

180 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 106 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)(18,19,23,30,25,31,49,62,26,32,48,29,24,52,22,47)(20,33,36,54,38,64,35,55,39,60,50,78,37,72,70,27)(21,40,43,74,45,59,42,61,46,75,51,79,44,28,56,58)(34,65,67,53,69,80,77,81,41,76,71,57,68,63,66,73);(18,20,22,35,24,37,48,36,26,39,49,70,25,38,23,50)(19,27,29,33,31,54,47,64,32,55,30,60,52,78,62,72)(21,41,42,77,44,69,43,67,46,34,56,66,45,68,51,71)(28,57,58,80,40,73,74,76,59,53,61,63,75,81,79,65);(18,21,24,44,26,46,25,45)(19,28,31,40,32,59,52,75)(20,34,37,68,39,41,38,69)(22,42,48,43,49,56,23,51)(27,53,54,81,55,57,78,73)(29,58,47,74,30,61,62,79)(33,63,64,65,60,80,72,76)(35,66,36,71,70,77,50,67)

Conjugacy classes & character table

884 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4352, 5015);