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G = C17×C7⋊C3order 357 = 3·7·17

Direct product of C17 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C17×C7⋊C3, C7⋊C51, C119⋊C3, SmallGroup(357,1)

Series: Derived Chief Lower central Upper central

C1C7 — C17×C7⋊C3
C1C7C119 — C17×C7⋊C3
C7 — C17×C7⋊C3
C1C17

Generators and relations for C17×C7⋊C3
 G = < a,b,c | a17=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C51

Smallest permutation representation of C17×C7⋊C3
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 31 114 58 85 102 39)(2 32 115 59 69 86 40)(3 33 116 60 70 87 41)(4 34 117 61 71 88 42)(5 18 118 62 72 89 43)(6 19 119 63 73 90 44)(7 20 103 64 74 91 45)(8 21 104 65 75 92 46)(9 22 105 66 76 93 47)(10 23 106 67 77 94 48)(11 24 107 68 78 95 49)(12 25 108 52 79 96 50)(13 26 109 53 80 97 51)(14 27 110 54 81 98 35)(15 28 111 55 82 99 36)(16 29 112 56 83 100 37)(17 30 113 57 84 101 38)
(18 118 72)(19 119 73)(20 103 74)(21 104 75)(22 105 76)(23 106 77)(24 107 78)(25 108 79)(26 109 80)(27 110 81)(28 111 82)(29 112 83)(30 113 84)(31 114 85)(32 115 69)(33 116 70)(34 117 71)(35 98 54)(36 99 55)(37 100 56)(38 101 57)(39 102 58)(40 86 59)(41 87 60)(42 88 61)(43 89 62)(44 90 63)(45 91 64)(46 92 65)(47 93 66)(48 94 67)(49 95 68)(50 96 52)(51 97 53)

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

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

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

85 conjugacy classes

class 1 3A3B7A7B17A···17P51A···51AF119A···119AF
order1337717···1751···51119···119
size177331···17···73···3

85 irreducible representations

dim111133
type+
imageC1C3C17C51C7⋊C3C17×C7⋊C3
kernelC17×C7⋊C3C119C7⋊C3C7C17C1
# reps121632232

Matrix representation of C17×C7⋊C3 in GL3(𝔽1429) generated by

115700
011570
001157
,
001
10502
01501
,
10501
001428
011428
G:=sub<GL(3,GF(1429))| [1157,0,0,0,1157,0,0,0,1157],[0,1,0,0,0,1,1,502,501],[1,0,0,0,0,1,501,1428,1428] >;

C17×C7⋊C3 in GAP, Magma, Sage, TeX

C_{17}\times C_7\rtimes C_3
% in TeX

G:=Group("C17xC7:C3");
// GroupNames label

G:=SmallGroup(357,1);
// by ID

G=gap.SmallGroup(357,1);
# by ID

G:=PCGroup([3,-3,-17,-7,920]);
// Polycyclic

G:=Group<a,b,c|a^17=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C17×C7⋊C3 in TeX

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