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G = D7×C17order 238 = 2·7·17

Direct product of C17 and D7

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

Aliases: D7×C17, C7⋊C34, C1193C2, SmallGroup(238,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C17
C1C7C119 — D7×C17
C7 — D7×C17
C1C17

Generators and relations for D7×C17
 G = < a,b,c | a17=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C34

Smallest permutation representation of D7×C17
On 119 points
Generators in S119
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)
(1 54 102 119 41 31 85)(2 55 86 103 42 32 69)(3 56 87 104 43 33 70)(4 57 88 105 44 34 71)(5 58 89 106 45 18 72)(6 59 90 107 46 19 73)(7 60 91 108 47 20 74)(8 61 92 109 48 21 75)(9 62 93 110 49 22 76)(10 63 94 111 50 23 77)(11 64 95 112 51 24 78)(12 65 96 113 35 25 79)(13 66 97 114 36 26 80)(14 67 98 115 37 27 81)(15 68 99 116 38 28 82)(16 52 100 117 39 29 83)(17 53 101 118 40 30 84)
(1 85)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)

G:=sub<Sym(119)| (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,54,102,119,41,31,85)(2,55,86,103,42,32,69)(3,56,87,104,43,33,70)(4,57,88,105,44,34,71)(5,58,89,106,45,18,72)(6,59,90,107,46,19,73)(7,60,91,108,47,20,74)(8,61,92,109,48,21,75)(9,62,93,110,49,22,76)(10,63,94,111,50,23,77)(11,64,95,112,51,24,78)(12,65,96,113,35,25,79)(13,66,97,114,36,26,80)(14,67,98,115,37,27,81)(15,68,99,116,38,28,82)(16,52,100,117,39,29,83)(17,53,101,118,40,30,84), (1,85)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95)>;

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)(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119), (1,54,102,119,41,31,85)(2,55,86,103,42,32,69)(3,56,87,104,43,33,70)(4,57,88,105,44,34,71)(5,58,89,106,45,18,72)(6,59,90,107,46,19,73)(7,60,91,108,47,20,74)(8,61,92,109,48,21,75)(9,62,93,110,49,22,76)(10,63,94,111,50,23,77)(11,64,95,112,51,24,78)(12,65,96,113,35,25,79)(13,66,97,114,36,26,80)(14,67,98,115,37,27,81)(15,68,99,116,38,28,82)(16,52,100,117,39,29,83)(17,53,101,118,40,30,84), (1,85)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,82)(16,83)(17,84)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,93)(50,94)(51,95) );

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),(86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)], [(1,54,102,119,41,31,85),(2,55,86,103,42,32,69),(3,56,87,104,43,33,70),(4,57,88,105,44,34,71),(5,58,89,106,45,18,72),(6,59,90,107,46,19,73),(7,60,91,108,47,20,74),(8,61,92,109,48,21,75),(9,62,93,110,49,22,76),(10,63,94,111,50,23,77),(11,64,95,112,51,24,78),(12,65,96,113,35,25,79),(13,66,97,114,36,26,80),(14,67,98,115,37,27,81),(15,68,99,116,38,28,82),(16,52,100,117,39,29,83),(17,53,101,118,40,30,84)], [(1,85),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,82),(16,83),(17,84),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92),(49,93),(50,94),(51,95)])

85 conjugacy classes

class 1  2 7A7B7C17A···17P34A···34P119A···119AV
order1277717···1734···34119···119
size172221···17···72···2

85 irreducible representations

dim111122
type+++
imageC1C2C17C34D7D7×C17
kernelD7×C17C119D7C7C17C1
# reps111616348

Matrix representation of D7×C17 in GL2(𝔽239) generated by

750
075
,
23249
23841
,
23248
2387
G:=sub<GL(2,GF(239))| [75,0,0,75],[232,238,49,41],[232,238,48,7] >;

D7×C17 in GAP, Magma, Sage, TeX

D_7\times C_{17}
% in TeX

G:=Group("D7xC17");
// GroupNames label

G:=SmallGroup(238,1);
// by ID

G=gap.SmallGroup(238,1);
# by ID

G:=PCGroup([3,-2,-17,-7,1838]);
// Polycyclic

G:=Group<a,b,c|a^17=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D7×C17 in TeX

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