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G = D5×C17order 170 = 2·5·17

Direct product of C17 and D5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D5×C17, C5⋊C34, C853C2, SmallGroup(170,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C17
C1C5C85 — D5×C17
C5 — D5×C17
C1C17

Generators and relations for D5×C17
 G = < a,b,c | a17=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C34

Smallest permutation representation of D5×C17
On 85 points
Generators in S85
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34)(35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51)(52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)(69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85)
(1 81 37 67 25)(2 82 38 68 26)(3 83 39 52 27)(4 84 40 53 28)(5 85 41 54 29)(6 69 42 55 30)(7 70 43 56 31)(8 71 44 57 32)(9 72 45 58 33)(10 73 46 59 34)(11 74 47 60 18)(12 75 48 61 19)(13 76 49 62 20)(14 77 50 63 21)(15 78 51 64 22)(16 79 35 65 23)(17 80 36 66 24)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(17 24)(52 83)(53 84)(54 85)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)

G:=sub<Sym(85)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85), (1,81,37,67,25)(2,82,38,68,26)(3,83,39,52,27)(4,84,40,53,28)(5,85,41,54,29)(6,69,42,55,30)(7,70,43,56,31)(8,71,44,57,32)(9,72,45,58,33)(10,73,46,59,34)(11,74,47,60,18)(12,75,48,61,19)(13,76,49,62,20)(14,77,50,63,21)(15,78,51,64,22)(16,79,35,65,23)(17,80,36,66,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(17,24)(52,83)(53,84)(54,85)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85), (1,81,37,67,25)(2,82,38,68,26)(3,83,39,52,27)(4,84,40,53,28)(5,85,41,54,29)(6,69,42,55,30)(7,70,43,56,31)(8,71,44,57,32)(9,72,45,58,33)(10,73,46,59,34)(11,74,47,60,18)(12,75,48,61,19)(13,76,49,62,20)(14,77,50,63,21)(15,78,51,64,22)(16,79,35,65,23)(17,80,36,66,24), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(17,24)(52,83)(53,84)(54,85)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85)], [(1,81,37,67,25),(2,82,38,68,26),(3,83,39,52,27),(4,84,40,53,28),(5,85,41,54,29),(6,69,42,55,30),(7,70,43,56,31),(8,71,44,57,32),(9,72,45,58,33),(10,73,46,59,34),(11,74,47,60,18),(12,75,48,61,19),(13,76,49,62,20),(14,77,50,63,21),(15,78,51,64,22),(16,79,35,65,23),(17,80,36,66,24)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23),(17,24),(52,83),(53,84),(54,85),(55,69),(56,70),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82)])

D5×C17 is a maximal subgroup of   C173F5

68 conjugacy classes

class 1  2 5A5B17A···17P34A···34P85A···85AF
order125517···1734···3485···85
size15221···15···52···2

68 irreducible representations

dim111122
type+++
imageC1C2C17C34D5D5×C17
kernelD5×C17C85D5C5C17C1
# reps111616232

Matrix representation of D5×C17 in GL2(𝔽1021) generated by

90
09
,
01
1020563
,
01
10
G:=sub<GL(2,GF(1021))| [9,0,0,9],[0,1020,1,563],[0,1,1,0] >;

D5×C17 in GAP, Magma, Sage, TeX

D_5\times C_{17}
% in TeX

G:=Group("D5xC17");
// GroupNames label

G:=SmallGroup(170,1);
// by ID

G=gap.SmallGroup(170,1);
# by ID

G:=PCGroup([3,-2,-17,-5,1226]);
// Polycyclic

G:=Group<a,b,c|a^17=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C17 in TeX

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