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G = C6×D17order 204 = 22·3·17

Direct product of C6 and D17

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

Aliases: C6×D17, C34⋊C6, C1022C2, C513C22, C17⋊(C2×C6), SmallGroup(204,9)

Series: Derived Chief Lower central Upper central

C1C17 — C6×D17
C1C17C51C3×D17 — C6×D17
C17 — C6×D17
C1C6

Generators and relations for C6×D17
 G = < a,b,c | a6=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C2
17C22
17C6
17C6
17C2×C6

Smallest permutation representation of C6×D17
On 102 points
Generators in S102
(1 78 46 66 23 91)(2 79 47 67 24 92)(3 80 48 68 25 93)(4 81 49 52 26 94)(5 82 50 53 27 95)(6 83 51 54 28 96)(7 84 35 55 29 97)(8 85 36 56 30 98)(9 69 37 57 31 99)(10 70 38 58 32 100)(11 71 39 59 33 101)(12 72 40 60 34 102)(13 73 41 61 18 86)(14 74 42 62 19 87)(15 75 43 63 20 88)(16 76 44 64 21 89)(17 77 45 65 22 90)
(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)(86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 68)(16 67)(17 66)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 72)(29 71)(30 70)(31 69)(32 85)(33 84)(34 83)(35 101)(36 100)(37 99)(38 98)(39 97)(40 96)(41 95)(42 94)(43 93)(44 92)(45 91)(46 90)(47 89)(48 88)(49 87)(50 86)(51 102)

G:=sub<Sym(102)| (1,78,46,66,23,91)(2,79,47,67,24,92)(3,80,48,68,25,93)(4,81,49,52,26,94)(5,82,50,53,27,95)(6,83,51,54,28,96)(7,84,35,55,29,97)(8,85,36,56,30,98)(9,69,37,57,31,99)(10,70,38,58,32,100)(11,71,39,59,33,101)(12,72,40,60,34,102)(13,73,41,61,18,86)(14,74,42,62,19,87)(15,75,43,63,20,88)(16,76,44,64,21,89)(17,77,45,65,22,90), (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)(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,68)(16,67)(17,66)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,85)(33,84)(34,83)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,102)>;

G:=Group( (1,78,46,66,23,91)(2,79,47,67,24,92)(3,80,48,68,25,93)(4,81,49,52,26,94)(5,82,50,53,27,95)(6,83,51,54,28,96)(7,84,35,55,29,97)(8,85,36,56,30,98)(9,69,37,57,31,99)(10,70,38,58,32,100)(11,71,39,59,33,101)(12,72,40,60,34,102)(13,73,41,61,18,86)(14,74,42,62,19,87)(15,75,43,63,20,88)(16,76,44,64,21,89)(17,77,45,65,22,90), (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)(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,68)(16,67)(17,66)(18,82)(19,81)(20,80)(21,79)(22,78)(23,77)(24,76)(25,75)(26,74)(27,73)(28,72)(29,71)(30,70)(31,69)(32,85)(33,84)(34,83)(35,101)(36,100)(37,99)(38,98)(39,97)(40,96)(41,95)(42,94)(43,93)(44,92)(45,91)(46,90)(47,89)(48,88)(49,87)(50,86)(51,102) );

G=PermutationGroup([(1,78,46,66,23,91),(2,79,47,67,24,92),(3,80,48,68,25,93),(4,81,49,52,26,94),(5,82,50,53,27,95),(6,83,51,54,28,96),(7,84,35,55,29,97),(8,85,36,56,30,98),(9,69,37,57,31,99),(10,70,38,58,32,100),(11,71,39,59,33,101),(12,72,40,60,34,102),(13,73,41,61,18,86),(14,74,42,62,19,87),(15,75,43,63,20,88),(16,76,44,64,21,89),(17,77,45,65,22,90)], [(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),(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,68),(16,67),(17,66),(18,82),(19,81),(20,80),(21,79),(22,78),(23,77),(24,76),(25,75),(26,74),(27,73),(28,72),(29,71),(30,70),(31,69),(32,85),(33,84),(34,83),(35,101),(36,100),(37,99),(38,98),(39,97),(40,96),(41,95),(42,94),(43,93),(44,92),(45,91),(46,90),(47,89),(48,88),(49,87),(50,86),(51,102)])

C6×D17 is a maximal subgroup of   C51⋊D4  C3⋊D68

60 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F17A···17H34A···34H51A···51P102A···102P
order12223366666617···1734···3451···51102···102
size1117171111171717172···22···22···22···2

60 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D17D34C3×D17C6×D17
kernelC6×D17C3×D17C102D34D17C34C6C3C2C1
# reps121242881616

Matrix representation of C6×D17 in GL3(𝔽103) generated by

10200
0460
0046
,
100
0971
06840
,
100
040102
05463
G:=sub<GL(3,GF(103))| [102,0,0,0,46,0,0,0,46],[1,0,0,0,97,68,0,1,40],[1,0,0,0,40,54,0,102,63] >;

C6×D17 in GAP, Magma, Sage, TeX

C_6\times D_{17}
% in TeX

G:=Group("C6xD17");
// GroupNames label

G:=SmallGroup(204,9);
// by ID

G=gap.SmallGroup(204,9);
# by ID

G:=PCGroup([4,-2,-2,-3,-17,3075]);
// Polycyclic

G:=Group<a,b,c|a^6=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D17 in TeX

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