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## G = C17×C7⋊C3order 357 = 3·7·17

### Direct product of C17 and C7⋊C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C17×C7⋊C3
 Chief series C1 — C7 — C119 — C17×C7⋊C3
 Lower central C7 — C17×C7⋊C3
 Upper central C1 — C17

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

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

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

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

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

85 conjugacy classes

 class 1 3A 3B 7A 7B 17A ··· 17P 51A ··· 51AF 119A ··· 119AF order 1 3 3 7 7 17 ··· 17 51 ··· 51 119 ··· 119 size 1 7 7 3 3 1 ··· 1 7 ··· 7 3 ··· 3

85 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C17 C51 C7⋊C3 C17×C7⋊C3 kernel C17×C7⋊C3 C119 C7⋊C3 C7 C17 C1 # reps 1 2 16 32 2 32

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

 1157 0 0 0 1157 0 0 0 1157
,
 0 0 1 1 0 502 0 1 501
,
 1 0 501 0 0 1428 0 1 1428
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

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