Copied to
clipboard

## G = C6×D17order 204 = 22·3·17

### Direct product of C6 and D17

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C6×D17
 Chief series C1 — C17 — C51 — C3×D17 — C6×D17
 Lower central C17 — C6×D17
 Upper central C1 — C6

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

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 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 17A ··· 17H 34A ··· 34H 51A ··· 51P 102A ··· 102P order 1 2 2 2 3 3 6 6 6 6 6 6 17 ··· 17 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 17 17 1 1 1 1 17 17 17 17 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D17 D34 C3×D17 C6×D17 kernel C6×D17 C3×D17 C102 D34 D17 C34 C6 C3 C2 C1 # reps 1 2 1 2 4 2 8 8 16 16

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

 102 0 0 0 46 0 0 0 46
,
 1 0 0 0 97 1 0 68 40
,
 1 0 0 0 40 102 0 54 63
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

׿
×
𝔽