direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C11×C13⋊C3, C143⋊C3, C13⋊C33, SmallGroup(429,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C11×C13⋊C3 |
Generators and relations for C11×C13⋊C3
G = < a,b,c | a11=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C11×C13⋊C3
►On 143 points
Generators in S143
(1 131 118 105 92 79 66 53 40 27 14)(2 132 119 106 93 80 67 54 41 28 15)(3 133 120 107 94 81 68 55 42 29 16)(4 134 121 108 95 82 69 56 43 30 17)(5 135 122 109 96 83 70 57 44 31 18)(6 136 123 110 97 84 71 58 45 32 19)(7 137 124 111 98 85 72 59 46 33 20)(8 138 125 112 99 86 73 60 47 34 21)(9 139 126 113 100 87 74 61 48 35 22)(10 140 127 114 101 88 75 62 49 36 23)(11 141 128 115 102 89 76 63 50 37 24)(12 142 129 116 103 90 77 64 51 38 25)(13 143 130 117 104 91 78 65 52 39 26)
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(15 17 23)(16 20 19)(18 26 24)(21 22 25)(28 30 36)(29 33 32)(31 39 37)(34 35 38)(41 43 49)(42 46 45)(44 52 50)(47 48 51)(54 56 62)(55 59 58)(57 65 63)(60 61 64)(67 69 75)(68 72 71)(70 78 76)(73 74 77)(80 82 88)(81 85 84)(83 91 89)(86 87 90)(93 95 101)(94 98 97)(96 104 102)(99 100 103)(106 108 114)(107 111 110)(109 117 115)(112 113 116)(119 121 127)(120 124 123)(122 130 128)(125 126 129)(132 134 140)(133 137 136)(135 143 141)(138 139 142)
G:=sub<Sym(143)| (1,131,118,105,92,79,66,53,40,27,14)(2,132,119,106,93,80,67,54,41,28,15)(3,133,120,107,94,81,68,55,42,29,16)(4,134,121,108,95,82,69,56,43,30,17)(5,135,122,109,96,83,70,57,44,31,18)(6,136,123,110,97,84,71,58,45,32,19)(7,137,124,111,98,85,72,59,46,33,20)(8,138,125,112,99,86,73,60,47,34,21)(9,139,126,113,100,87,74,61,48,35,22)(10,140,127,114,101,88,75,62,49,36,23)(11,141,128,115,102,89,76,63,50,37,24)(12,142,129,116,103,90,77,64,51,38,25)(13,143,130,117,104,91,78,65,52,39,26), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(119,121,127)(120,124,123)(122,130,128)(125,126,129)(132,134,140)(133,137,136)(135,143,141)(138,139,142)>;
G:=Group( (1,131,118,105,92,79,66,53,40,27,14)(2,132,119,106,93,80,67,54,41,28,15)(3,133,120,107,94,81,68,55,42,29,16)(4,134,121,108,95,82,69,56,43,30,17)(5,135,122,109,96,83,70,57,44,31,18)(6,136,123,110,97,84,71,58,45,32,19)(7,137,124,111,98,85,72,59,46,33,20)(8,138,125,112,99,86,73,60,47,34,21)(9,139,126,113,100,87,74,61,48,35,22)(10,140,127,114,101,88,75,62,49,36,23)(11,141,128,115,102,89,76,63,50,37,24)(12,142,129,116,103,90,77,64,51,38,25)(13,143,130,117,104,91,78,65,52,39,26), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143), (2,4,10)(3,7,6)(5,13,11)(8,9,12)(15,17,23)(16,20,19)(18,26,24)(21,22,25)(28,30,36)(29,33,32)(31,39,37)(34,35,38)(41,43,49)(42,46,45)(44,52,50)(47,48,51)(54,56,62)(55,59,58)(57,65,63)(60,61,64)(67,69,75)(68,72,71)(70,78,76)(73,74,77)(80,82,88)(81,85,84)(83,91,89)(86,87,90)(93,95,101)(94,98,97)(96,104,102)(99,100,103)(106,108,114)(107,111,110)(109,117,115)(112,113,116)(119,121,127)(120,124,123)(122,130,128)(125,126,129)(132,134,140)(133,137,136)(135,143,141)(138,139,142) );
G=PermutationGroup([[(1,131,118,105,92,79,66,53,40,27,14),(2,132,119,106,93,80,67,54,41,28,15),(3,133,120,107,94,81,68,55,42,29,16),(4,134,121,108,95,82,69,56,43,30,17),(5,135,122,109,96,83,70,57,44,31,18),(6,136,123,110,97,84,71,58,45,32,19),(7,137,124,111,98,85,72,59,46,33,20),(8,138,125,112,99,86,73,60,47,34,21),(9,139,126,113,100,87,74,61,48,35,22),(10,140,127,114,101,88,75,62,49,36,23),(11,141,128,115,102,89,76,63,50,37,24),(12,142,129,116,103,90,77,64,51,38,25),(13,143,130,117,104,91,78,65,52,39,26)], [(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),(131,132,133,134,135,136,137,138,139,140,141,142,143)], [(2,4,10),(3,7,6),(5,13,11),(8,9,12),(15,17,23),(16,20,19),(18,26,24),(21,22,25),(28,30,36),(29,33,32),(31,39,37),(34,35,38),(41,43,49),(42,46,45),(44,52,50),(47,48,51),(54,56,62),(55,59,58),(57,65,63),(60,61,64),(67,69,75),(68,72,71),(70,78,76),(73,74,77),(80,82,88),(81,85,84),(83,91,89),(86,87,90),(93,95,101),(94,98,97),(96,104,102),(99,100,103),(106,108,114),(107,111,110),(109,117,115),(112,113,116),(119,121,127),(120,124,123),(122,130,128),(125,126,129),(132,134,140),(133,137,136),(135,143,141),(138,139,142)]])
77 conjugacy classes
| class | 1 | 3A | 3B | 11A | ··· | 11J | 13A | 13B | 13C | 13D | 33A | ··· | 33T | 143A | ··· | 143AN |
| order | 1 | 3 | 3 | 11 | ··· | 11 | 13 | 13 | 13 | 13 | 33 | ··· | 33 | 143 | ··· | 143 |
| size | 1 | 13 | 13 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | ··· | 3 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C11 | C33 | C13⋊C3 | C11×C13⋊C3 |
| kernel | C11×C13⋊C3 | C143 | C13⋊C3 | C13 | C11 | C1 |
| # reps | 1 | 2 | 10 | 20 | 4 | 40 |
Matrix representation of C11×C13⋊C3 ►in GL3(𝔽859) generated by
| 479 | 0 | 0 |
| 0 | 479 | 0 |
| 0 | 0 | 479 |
| 0 | 0 | 1 |
| 1 | 0 | 455 |
| 0 | 1 | 69 |
| 1 | 1 | 455 |
| 0 | 455 | 68 |
| 0 | 69 | 403 |
G:=sub<GL(3,GF(859))| [479,0,0,0,479,0,0,0,479],[0,1,0,0,0,1,1,455,69],[1,0,0,1,455,69,455,68,403] >;
C11×C13⋊C3 in GAP, Magma, Sage, TeX
C_{11}\times C_{13}\rtimes C_3 % in TeX
G:=Group("C11xC13:C3"); // GroupNames label
G:=SmallGroup(429,1);
// by ID
G=gap.SmallGroup(429,1);
# by ID
G:=PCGroup([3,-3,-11,-13,893]);
// Polycyclic
G:=Group<a,b,c|a^11=b^13=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^9>;
// generators/relations
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