direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C10×C13⋊C3, C130⋊C3, C26⋊C15, C65⋊4C6, C13⋊2C30, SmallGroup(390,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C10×C13⋊C3 |
Generators and relations for C10×C13⋊C3
G = < a,b,c | a10=b13=c3=1, ab=ba, ac=ca, cbc-1=b9 >
Smallest permutation representation of C10×C13⋊C3
►On 130 points
Generators in S130
(1 92 53 79 40 66 27 118 14 105)(2 93 54 80 41 67 28 119 15 106)(3 94 55 81 42 68 29 120 16 107)(4 95 56 82 43 69 30 121 17 108)(5 96 57 83 44 70 31 122 18 109)(6 97 58 84 45 71 32 123 19 110)(7 98 59 85 46 72 33 124 20 111)(8 99 60 86 47 73 34 125 21 112)(9 100 61 87 48 74 35 126 22 113)(10 101 62 88 49 75 36 127 23 114)(11 102 63 89 50 76 37 128 24 115)(12 103 64 90 51 77 38 129 25 116)(13 104 65 91 52 78 39 130 26 117)
(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)
(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)
G:=sub<Sym(130)| (1,92,53,79,40,66,27,118,14,105)(2,93,54,80,41,67,28,119,15,106)(3,94,55,81,42,68,29,120,16,107)(4,95,56,82,43,69,30,121,17,108)(5,96,57,83,44,70,31,122,18,109)(6,97,58,84,45,71,32,123,19,110)(7,98,59,85,46,72,33,124,20,111)(8,99,60,86,47,73,34,125,21,112)(9,100,61,87,48,74,35,126,22,113)(10,101,62,88,49,75,36,127,23,114)(11,102,63,89,50,76,37,128,24,115)(12,103,64,90,51,77,38,129,25,116)(13,104,65,91,52,78,39,130,26,117), (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), (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)>;
G:=Group( (1,92,53,79,40,66,27,118,14,105)(2,93,54,80,41,67,28,119,15,106)(3,94,55,81,42,68,29,120,16,107)(4,95,56,82,43,69,30,121,17,108)(5,96,57,83,44,70,31,122,18,109)(6,97,58,84,45,71,32,123,19,110)(7,98,59,85,46,72,33,124,20,111)(8,99,60,86,47,73,34,125,21,112)(9,100,61,87,48,74,35,126,22,113)(10,101,62,88,49,75,36,127,23,114)(11,102,63,89,50,76,37,128,24,115)(12,103,64,90,51,77,38,129,25,116)(13,104,65,91,52,78,39,130,26,117), (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), (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) );
G=PermutationGroup([[(1,92,53,79,40,66,27,118,14,105),(2,93,54,80,41,67,28,119,15,106),(3,94,55,81,42,68,29,120,16,107),(4,95,56,82,43,69,30,121,17,108),(5,96,57,83,44,70,31,122,18,109),(6,97,58,84,45,71,32,123,19,110),(7,98,59,85,46,72,33,124,20,111),(8,99,60,86,47,73,34,125,21,112),(9,100,61,87,48,74,35,126,22,113),(10,101,62,88,49,75,36,127,23,114),(11,102,63,89,50,76,37,128,24,115),(12,103,64,90,51,77,38,129,25,116),(13,104,65,91,52,78,39,130,26,117)], [(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)], [(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)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 13A | 13B | 13C | 13D | 15A | ··· | 15H | 26A | 26B | 26C | 26D | 30A | ··· | 30H | 65A | ··· | 65P | 130A | ··· | 130P |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 13 | 13 | 13 | 13 | 15 | ··· | 15 | 26 | 26 | 26 | 26 | 30 | ··· | 30 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | 3 | 3 | 3 | 13 | ··· | 13 | 3 | ··· | 3 | 3 | ··· | 3 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | C13⋊C3 | C2×C13⋊C3 | C5×C13⋊C3 | C10×C13⋊C3 |
| kernel | C10×C13⋊C3 | C5×C13⋊C3 | C130 | C2×C13⋊C3 | C65 | C13⋊C3 | C26 | C13 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 4 | 16 | 16 |
Matrix representation of C10×C13⋊C3 ►in GL3(𝔽1171) generated by
| 955 | 0 | 0 |
| 0 | 955 | 0 |
| 0 | 0 | 955 |
| 760 | 21 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1149 | 759 | 21 |
| 852 | 22 | 411 |
G:=sub<GL(3,GF(1171))| [955,0,0,0,955,0,0,0,955],[760,1,0,21,0,1,1,0,0],[1,1149,852,0,759,22,0,21,411] >;
C10×C13⋊C3 in GAP, Magma, Sage, TeX
C_{10}\times C_{13}\rtimes C_3 % in TeX
G:=Group("C10xC13:C3"); // GroupNames label
G:=SmallGroup(390,4);
// by ID
G=gap.SmallGroup(390,4);
# by ID
G:=PCGroup([4,-2,-3,-5,-13,727]);
// Polycyclic
G:=Group<a,b,c|a^10=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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