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G = C5×D17order 170 = 2·5·17

Direct product of C5 and D17

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

Aliases: C5×D17, C17⋊C10, C852C2, SmallGroup(170,2)

Series: Derived Chief Lower central Upper central

C1C17 — C5×D17
C1C17C85 — C5×D17
C17 — C5×D17
C1C5

Generators and relations for C5×D17
 G = < a,b,c | a5=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C10

Smallest permutation representation of C5×D17
On 85 points
Generators in S85
(1 69 68 45 29)(2 70 52 46 30)(3 71 53 47 31)(4 72 54 48 32)(5 73 55 49 33)(6 74 56 50 34)(7 75 57 51 18)(8 76 58 35 19)(9 77 59 36 20)(10 78 60 37 21)(11 79 61 38 22)(12 80 62 39 23)(13 81 63 40 24)(14 82 64 41 25)(15 83 65 42 26)(16 84 66 43 27)(17 85 67 44 28)
(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 22)(19 21)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(35 37)(38 51)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(67 68)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)

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

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

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

C5×D17 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+++
imageC1C2C5C10D17C5×D17
kernelC5×D17C85D17C17C5C1
# reps1144832

Matrix representation of C5×D17 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] >;

C5×D17 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 C5×D17 in TeX

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