direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: C5×C7⋊C12, C7⋊C60, C35⋊4C12, C14.C30, Dic7⋊C15, C70.2C6, C10.2F7, C7⋊C3⋊C20, C2.(C5×F7), (C5×Dic7)⋊C3, (C5×C7⋊C3)⋊4C4, (C2×C7⋊C3).C10, (C10×C7⋊C3).2C2, SmallGroup(420,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C5×C7⋊C12 |
Generators and relations for C5×C7⋊C12
G = < a,b,c | a5=b7=c12=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C5×C7⋊C12
►On 140 points
Generators in S140
(1 8 10 19 16)(2 5 11 20 13)(3 6 12 17 14)(4 7 9 18 15)(21 126 60 56 35)(22 127 61 45 36)(23 128 62 46 37)(24 117 63 47 38)(25 118 64 48 39)(26 119 65 49 40)(27 120 66 50 41)(28 121 67 51 42)(29 122 68 52 43)(30 123 57 53 44)(31 124 58 54 33)(32 125 59 55 34)(69 93 137 112 86)(70 94 138 113 87)(71 95 139 114 88)(72 96 140 115 89)(73 97 129 116 90)(74 98 130 105 91)(75 99 131 106 92)(76 100 132 107 81)(77 101 133 108 82)(78 102 134 109 83)(79 103 135 110 84)(80 104 136 111 85)
(1 138 134 28 130 32 24)(2 21 29 139 25 131 135)(3 132 140 22 136 26 30)(4 27 23 133 31 137 129)(5 126 122 114 118 106 110)(6 107 115 127 111 119 123)(7 120 128 108 124 112 116)(8 113 109 121 105 125 117)(9 66 62 82 58 86 90)(10 87 83 67 91 59 63)(11 60 68 88 64 92 84)(12 81 89 61 85 65 57)(13 35 43 95 39 99 103)(14 100 96 36 104 40 44)(15 41 37 101 33 93 97)(16 94 102 42 98 34 38)(17 76 72 45 80 49 53)(18 50 46 77 54 69 73)(19 70 78 51 74 55 47)(20 56 52 71 48 75 79)
(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)(117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140)
G:=sub<Sym(140)| (1,8,10,19,16)(2,5,11,20,13)(3,6,12,17,14)(4,7,9,18,15)(21,126,60,56,35)(22,127,61,45,36)(23,128,62,46,37)(24,117,63,47,38)(25,118,64,48,39)(26,119,65,49,40)(27,120,66,50,41)(28,121,67,51,42)(29,122,68,52,43)(30,123,57,53,44)(31,124,58,54,33)(32,125,59,55,34)(69,93,137,112,86)(70,94,138,113,87)(71,95,139,114,88)(72,96,140,115,89)(73,97,129,116,90)(74,98,130,105,91)(75,99,131,106,92)(76,100,132,107,81)(77,101,133,108,82)(78,102,134,109,83)(79,103,135,110,84)(80,104,136,111,85), (1,138,134,28,130,32,24)(2,21,29,139,25,131,135)(3,132,140,22,136,26,30)(4,27,23,133,31,137,129)(5,126,122,114,118,106,110)(6,107,115,127,111,119,123)(7,120,128,108,124,112,116)(8,113,109,121,105,125,117)(9,66,62,82,58,86,90)(10,87,83,67,91,59,63)(11,60,68,88,64,92,84)(12,81,89,61,85,65,57)(13,35,43,95,39,99,103)(14,100,96,36,104,40,44)(15,41,37,101,33,93,97)(16,94,102,42,98,34,38)(17,76,72,45,80,49,53)(18,50,46,77,54,69,73)(19,70,78,51,74,55,47)(20,56,52,71,48,75,79), (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)(117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140)>;
G:=Group( (1,8,10,19,16)(2,5,11,20,13)(3,6,12,17,14)(4,7,9,18,15)(21,126,60,56,35)(22,127,61,45,36)(23,128,62,46,37)(24,117,63,47,38)(25,118,64,48,39)(26,119,65,49,40)(27,120,66,50,41)(28,121,67,51,42)(29,122,68,52,43)(30,123,57,53,44)(31,124,58,54,33)(32,125,59,55,34)(69,93,137,112,86)(70,94,138,113,87)(71,95,139,114,88)(72,96,140,115,89)(73,97,129,116,90)(74,98,130,105,91)(75,99,131,106,92)(76,100,132,107,81)(77,101,133,108,82)(78,102,134,109,83)(79,103,135,110,84)(80,104,136,111,85), (1,138,134,28,130,32,24)(2,21,29,139,25,131,135)(3,132,140,22,136,26,30)(4,27,23,133,31,137,129)(5,126,122,114,118,106,110)(6,107,115,127,111,119,123)(7,120,128,108,124,112,116)(8,113,109,121,105,125,117)(9,66,62,82,58,86,90)(10,87,83,67,91,59,63)(11,60,68,88,64,92,84)(12,81,89,61,85,65,57)(13,35,43,95,39,99,103)(14,100,96,36,104,40,44)(15,41,37,101,33,93,97)(16,94,102,42,98,34,38)(17,76,72,45,80,49,53)(18,50,46,77,54,69,73)(19,70,78,51,74,55,47)(20,56,52,71,48,75,79), (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)(117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140) );
G=PermutationGroup([[(1,8,10,19,16),(2,5,11,20,13),(3,6,12,17,14),(4,7,9,18,15),(21,126,60,56,35),(22,127,61,45,36),(23,128,62,46,37),(24,117,63,47,38),(25,118,64,48,39),(26,119,65,49,40),(27,120,66,50,41),(28,121,67,51,42),(29,122,68,52,43),(30,123,57,53,44),(31,124,58,54,33),(32,125,59,55,34),(69,93,137,112,86),(70,94,138,113,87),(71,95,139,114,88),(72,96,140,115,89),(73,97,129,116,90),(74,98,130,105,91),(75,99,131,106,92),(76,100,132,107,81),(77,101,133,108,82),(78,102,134,109,83),(79,103,135,110,84),(80,104,136,111,85)], [(1,138,134,28,130,32,24),(2,21,29,139,25,131,135),(3,132,140,22,136,26,30),(4,27,23,133,31,137,129),(5,126,122,114,118,106,110),(6,107,115,127,111,119,123),(7,120,128,108,124,112,116),(8,113,109,121,105,125,117),(9,66,62,82,58,86,90),(10,87,83,67,91,59,63),(11,60,68,88,64,92,84),(12,81,89,61,85,65,57),(13,35,43,95,39,99,103),(14,100,96,36,104,40,44),(15,41,37,101,33,93,97),(16,94,102,42,98,34,38),(17,76,72,45,80,49,53),(18,50,46,77,54,69,73),(19,70,78,51,74,55,47),(20,56,52,71,48,75,79)], [(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),(117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 7 | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 14 | 15A | ··· | 15H | 20A | ··· | 20H | 30A | ··· | 30H | 35A | 35B | 35C | 35D | 60A | ··· | 60P | 70A | 70B | 70C | 70D |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 7 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 14 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 35 | 35 | 35 | 35 | 60 | ··· | 60 | 70 | 70 | 70 | 70 |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | 7 | 6 | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | 6 | 6 | 6 | 7 | ··· | 7 | 6 | 6 | 6 | 6 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | - | ||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | F7 | C7⋊C12 | C5×F7 | C5×C7⋊C12 |
| kernel | C5×C7⋊C12 | C10×C7⋊C3 | C5×Dic7 | C5×C7⋊C3 | C7⋊C12 | C70 | C2×C7⋊C3 | C35 | Dic7 | C7⋊C3 | C14 | C7 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C7⋊C12 ►in GL7(𝔽421)
| 279 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 377 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 377 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 377 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 377 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 377 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 377 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 420 |
| 0 | 1 | 0 | 0 | 0 | 0 | 420 |
| 0 | 0 | 1 | 0 | 0 | 0 | 420 |
| 0 | 0 | 0 | 1 | 0 | 0 | 420 |
| 0 | 0 | 0 | 0 | 1 | 0 | 420 |
| 0 | 0 | 0 | 0 | 0 | 1 | 420 |
| 262 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 201 | 0 | 0 | 201 | 199 | 220 |
| 0 | 201 | 201 | 199 | 0 | 220 | 0 |
| 0 | 400 | 0 | 220 | 201 | 220 | 0 |
| 0 | 0 | 201 | 220 | 201 | 0 | 199 |
| 0 | 0 | 201 | 0 | 400 | 220 | 220 |
| 0 | 201 | 400 | 220 | 0 | 0 | 220 |
G:=sub<GL(7,GF(421))| [279,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377,0,0,0,0,0,0,0,377],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,420,420,420,420,420,420],[262,0,0,0,0,0,0,0,201,201,400,0,0,201,0,0,201,0,201,201,400,0,0,199,220,220,0,220,0,201,0,201,201,400,0,0,199,220,220,0,220,0,0,220,0,0,199,220,220] >;
C5×C7⋊C12 in GAP, Magma, Sage, TeX
C_5\times C_7\rtimes C_{12} % in TeX
G:=Group("C5xC7:C12"); // GroupNames label
G:=SmallGroup(420,1);
// by ID
G=gap.SmallGroup(420,1);
# by ID
G:=PCGroup([5,-2,-3,-5,-2,-7,150,9004,3009]);
// Polycyclic
G:=Group<a,b,c|a^5=b^7=c^12=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
Export