C₁₇×#8515 #16866259

order	4352
nilpotency class	3
isoclinism	isoclinic class of #8510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₇×#8515

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₇	#379
commutator	C₄²	C₂²×C₄×C₁₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₇
prime radical	C₁₇×#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₂×C₄²×C₁₇ ◁ G

644 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 138 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)(18,19,22,29,25,20,31,35)(21,37,40,73,42,63,59,72)(23,45,43,55,48,71,60,66)(24,30,44,27,49,36,61,33)(26,50,52,77,32,62,51,39)(28,56,58,68,34,67,70,47)(38,74,41,57,76,81,78,69)(46,53,79,54,80,64,75,65);(18,20)(19,25)(21,32)(22,27)(23,45)(24,36)(26,42)(28,47)(29,44)(30,49)(31,33)(34,68)(35,61)(37,46)(38,51)(39,57)(40,54)(41,64)(43,70)(48,71)(50,72)(52,76)(53,78)(55,67)(56,66)(58,60)(59,65)(62,73)(63,80)(69,77)(74,79)(75,81);(18,21,24,41)(19,26,30,53)(20,32,36,64)(22,40,44,76)(23,46,47,39)(25,42,49,78)(27,54,29,52)(28,57,45,37)(31,59,61,38)(33,65,35,51)(34,69,71,63)(43,79,56,50)(48,80,68,77)(55,73,58,81)(60,75,67,62)(66,72,70,74)

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, 1503);