metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C133⋊3C3, C7⋊1(C19⋊C3), C19⋊1(C7⋊C3), SmallGroup(399,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C133 — C133⋊C3 |
Generators and relations for C133⋊C3
G = < a,b | a133=b3=1, bab-1=a11 >
Smallest permutation representation of C133⋊C3
►On 133 points
Generators in S133
(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)
(2 122 12)(3 110 23)(4 98 34)(5 86 45)(6 74 56)(7 62 67)(8 50 78)(9 38 89)(10 26 100)(11 14 111)(13 123 133)(15 99 22)(16 87 33)(17 75 44)(18 63 55)(19 51 66)(20 39 77)(21 27 88)(24 124 121)(25 112 132)(28 76 32)(29 64 43)(30 52 54)(31 40 65)(35 125 109)(36 113 120)(37 101 131)(41 53 42)(46 126 97)(47 114 108)(48 102 119)(49 90 130)(57 127 85)(58 115 96)(59 103 107)(60 91 118)(61 79 129)(68 128 73)(69 116 84)(70 104 95)(71 92 106)(72 80 117)(81 105 83)(82 93 94)
G:=sub<Sym(133)| (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), (2,122,12)(3,110,23)(4,98,34)(5,86,45)(6,74,56)(7,62,67)(8,50,78)(9,38,89)(10,26,100)(11,14,111)(13,123,133)(15,99,22)(16,87,33)(17,75,44)(18,63,55)(19,51,66)(20,39,77)(21,27,88)(24,124,121)(25,112,132)(28,76,32)(29,64,43)(30,52,54)(31,40,65)(35,125,109)(36,113,120)(37,101,131)(41,53,42)(46,126,97)(47,114,108)(48,102,119)(49,90,130)(57,127,85)(58,115,96)(59,103,107)(60,91,118)(61,79,129)(68,128,73)(69,116,84)(70,104,95)(71,92,106)(72,80,117)(81,105,83)(82,93,94)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133), (2,122,12)(3,110,23)(4,98,34)(5,86,45)(6,74,56)(7,62,67)(8,50,78)(9,38,89)(10,26,100)(11,14,111)(13,123,133)(15,99,22)(16,87,33)(17,75,44)(18,63,55)(19,51,66)(20,39,77)(21,27,88)(24,124,121)(25,112,132)(28,76,32)(29,64,43)(30,52,54)(31,40,65)(35,125,109)(36,113,120)(37,101,131)(41,53,42)(46,126,97)(47,114,108)(48,102,119)(49,90,130)(57,127,85)(58,115,96)(59,103,107)(60,91,118)(61,79,129)(68,128,73)(69,116,84)(70,104,95)(71,92,106)(72,80,117)(81,105,83)(82,93,94) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)], [(2,122,12),(3,110,23),(4,98,34),(5,86,45),(6,74,56),(7,62,67),(8,50,78),(9,38,89),(10,26,100),(11,14,111),(13,123,133),(15,99,22),(16,87,33),(17,75,44),(18,63,55),(19,51,66),(20,39,77),(21,27,88),(24,124,121),(25,112,132),(28,76,32),(29,64,43),(30,52,54),(31,40,65),(35,125,109),(36,113,120),(37,101,131),(41,53,42),(46,126,97),(47,114,108),(48,102,119),(49,90,130),(57,127,85),(58,115,96),(59,103,107),(60,91,118),(61,79,129),(68,128,73),(69,116,84),(70,104,95),(71,92,106),(72,80,117),(81,105,83),(82,93,94)]])
47 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 19A | ··· | 19F | 133A | ··· | 133AJ |
| order | 1 | 3 | 3 | 7 | 7 | 19 | ··· | 19 | 133 | ··· | 133 |
| size | 1 | 133 | 133 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
47 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||
| image | C1 | C3 | C7⋊C3 | C19⋊C3 | C133⋊C3 |
| kernel | C133⋊C3 | C133 | C19 | C7 | C1 |
| # reps | 1 | 2 | 2 | 6 | 36 |
Matrix representation of C133⋊C3 ►in GL3(𝔽11) generated by
| 8 | 0 | 4 |
| 4 | 0 | 6 |
| 6 | 1 | 3 |
| 1 | 2 | 0 |
| 0 | 10 | 1 |
| 0 | 10 | 0 |
G:=sub<GL(3,GF(11))| [8,4,6,0,0,1,4,6,3],[1,0,0,2,10,10,0,1,0] >;
C133⋊C3 in GAP, Magma, Sage, TeX
C_{133}\rtimes C_3 % in TeX
G:=Group("C133:C3"); // GroupNames label
G:=SmallGroup(399,3);
// by ID
G=gap.SmallGroup(399,3);
# by ID
G:=PCGroup([3,-3,-7,-19,37,1325]);
// Polycyclic
G:=Group<a,b|a^133=b^3=1,b*a*b^-1=a^11>;
// generators/relations
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