metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C133⋊4C3, C7⋊2(C19⋊C3), C19⋊2(C7⋊C3), SmallGroup(399,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C133 — C133⋊4C3 |
Generators and relations for C133⋊4C3
G = < a,b | a133=b3=1, bab-1=a30 >
Smallest permutation representation of C133⋊4C3
►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)
(2 103 31)(3 72 61)(4 41 91)(5 10 121)(6 112 18)(7 81 48)(8 50 78)(9 19 108)(11 90 35)(12 59 65)(13 28 95)(14 130 125)(15 99 22)(16 68 52)(17 37 82)(20 77 39)(21 46 69)(23 117 129)(24 86 26)(25 55 56)(27 126 116)(29 64 43)(30 33 73)(32 104 133)(34 42 60)(36 113 120)(38 51 47)(40 122 107)(44 131 94)(45 100 124)(49 109 111)(53 118 98)(54 87 128)(57 127 85)(58 96 115)(62 105 102)(63 74 132)(66 114 89)(67 83 119)(70 123 76)(71 92 106)(75 101 93)(79 110 80)(84 88 97)
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), (2,103,31)(3,72,61)(4,41,91)(5,10,121)(6,112,18)(7,81,48)(8,50,78)(9,19,108)(11,90,35)(12,59,65)(13,28,95)(14,130,125)(15,99,22)(16,68,52)(17,37,82)(20,77,39)(21,46,69)(23,117,129)(24,86,26)(25,55,56)(27,126,116)(29,64,43)(30,33,73)(32,104,133)(34,42,60)(36,113,120)(38,51,47)(40,122,107)(44,131,94)(45,100,124)(49,109,111)(53,118,98)(54,87,128)(57,127,85)(58,96,115)(62,105,102)(63,74,132)(66,114,89)(67,83,119)(70,123,76)(71,92,106)(75,101,93)(79,110,80)(84,88,97)>;
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), (2,103,31)(3,72,61)(4,41,91)(5,10,121)(6,112,18)(7,81,48)(8,50,78)(9,19,108)(11,90,35)(12,59,65)(13,28,95)(14,130,125)(15,99,22)(16,68,52)(17,37,82)(20,77,39)(21,46,69)(23,117,129)(24,86,26)(25,55,56)(27,126,116)(29,64,43)(30,33,73)(32,104,133)(34,42,60)(36,113,120)(38,51,47)(40,122,107)(44,131,94)(45,100,124)(49,109,111)(53,118,98)(54,87,128)(57,127,85)(58,96,115)(62,105,102)(63,74,132)(66,114,89)(67,83,119)(70,123,76)(71,92,106)(75,101,93)(79,110,80)(84,88,97) );
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)], [(2,103,31),(3,72,61),(4,41,91),(5,10,121),(6,112,18),(7,81,48),(8,50,78),(9,19,108),(11,90,35),(12,59,65),(13,28,95),(14,130,125),(15,99,22),(16,68,52),(17,37,82),(20,77,39),(21,46,69),(23,117,129),(24,86,26),(25,55,56),(27,126,116),(29,64,43),(30,33,73),(32,104,133),(34,42,60),(36,113,120),(38,51,47),(40,122,107),(44,131,94),(45,100,124),(49,109,111),(53,118,98),(54,87,128),(57,127,85),(58,96,115),(62,105,102),(63,74,132),(66,114,89),(67,83,119),(70,123,76),(71,92,106),(75,101,93),(79,110,80),(84,88,97)]])
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⋊4C3 |
| kernel | C133⋊4C3 | C133 | C19 | C7 | C1 |
| # reps | 1 | 2 | 2 | 6 | 36 |
Matrix representation of C133⋊4C3 ►in GL3(𝔽1597) generated by
| 153 | 669 | 830 |
| 830 | 1242 | 397 |
| 397 | 410 | 577 |
| 1 | 0 | 0 |
| 1352 | 1123 | 1414 |
| 1 | 719 | 473 |
G:=sub<GL(3,GF(1597))| [153,830,397,669,1242,410,830,397,577],[1,1352,1,0,1123,719,0,1414,473] >;
C133⋊4C3 in GAP, Magma, Sage, TeX
C_{133}\rtimes_4C_3 % in TeX
G:=Group("C133:4C3"); // GroupNames label
G:=SmallGroup(399,4);
// by ID
G=gap.SmallGroup(399,4);
# by ID
G:=PCGroup([3,-3,-7,-19,73,1325]);
// Polycyclic
G:=Group<a,b|a^133=b^3=1,b*a*b^-1=a^30>;
// generators/relations
Export