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G = C5xD17order 170 = 2·5·17

Direct product of C5 and D17

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

Aliases: C5xD17, C17:C10, C85:2C2, SmallGroup(170,2)

Series: Derived Chief Lower central Upper central

C1C17 — C5xD17
C1C17C85 — C5xD17
C17 — C5xD17
C1C5

Generators and relations for C5xD17
 G = < a,b,c | a5=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 40 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C5, C10, D17, C5xD17
17C2
17C10

Smallest permutation representation of C5xD17
On 85 points
Generators in S85
(1 76 56 42 32)(2 77 57 43 33)(3 78 58 44 34)(4 79 59 45 18)(5 80 60 46 19)(6 81 61 47 20)(7 82 62 48 21)(8 83 63 49 22)(9 84 64 50 23)(10 85 65 51 24)(11 69 66 35 25)(12 70 67 36 26)(13 71 68 37 27)(14 72 52 38 28)(15 73 53 39 29)(16 74 54 40 30)(17 75 55 41 31)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(83 85)

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

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

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

C5xD17 is a maximal subgroup of   C85:C4

50 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D17A···17H85A···85AF
order1255551010101017···1785···85
size1171111171717172···22···2

50 irreducible representations

dim111122
type+++
imageC1C2C5C10D17C5xD17
kernelC5xD17C85D17C17C5C1
# reps1144832

Matrix representation of C5xD17 in GL2(F1021) generated by

6760
0676
,
01
1020151
,
01
10
G:=sub<GL(2,GF(1021))| [676,0,0,676],[0,1020,1,151],[0,1,1,0] >;

C5xD17 in GAP, Magma, Sage, TeX

C_5\times D_{17}
% in TeX

G:=Group("C5xD17");
// GroupNames label

G:=SmallGroup(170,2);
// by ID

G=gap.SmallGroup(170,2);
# by ID

G:=PCGroup([3,-2,-5,-17,1442]);
// Polycyclic

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

Export

Subgroup lattice of C5xD17 in TeX

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