metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C139⋊C3, SmallGroup(417,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C139 — C139⋊C3 |
Generators and relations for C139⋊C3
G = < a,b | a139=b3=1, bab-1=a42 >
Smallest permutation representation of C139⋊C3
►On 139 points: primitive
Generators in S139
(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)
(2 97 43)(3 54 85)(4 11 127)(5 107 30)(6 64 72)(7 21 114)(8 117 17)(9 74 59)(10 31 101)(12 84 46)(13 41 88)(14 137 130)(15 94 33)(16 51 75)(18 104 20)(19 61 62)(22 71 49)(23 28 91)(24 124 133)(25 81 36)(26 38 78)(27 134 120)(29 48 65)(32 58 52)(34 111 136)(35 68 39)(37 121 123)(40 131 110)(42 45 55)(44 98 139)(47 108 126)(50 118 113)(53 128 100)(56 138 87)(57 95 129)(60 105 116)(63 115 103)(66 125 90)(67 82 132)(69 135 77)(70 92 119)(73 102 106)(76 112 93)(79 122 80)(83 89 109)(86 99 96)
G:=sub<Sym(139)| (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), (2,97,43)(3,54,85)(4,11,127)(5,107,30)(6,64,72)(7,21,114)(8,117,17)(9,74,59)(10,31,101)(12,84,46)(13,41,88)(14,137,130)(15,94,33)(16,51,75)(18,104,20)(19,61,62)(22,71,49)(23,28,91)(24,124,133)(25,81,36)(26,38,78)(27,134,120)(29,48,65)(32,58,52)(34,111,136)(35,68,39)(37,121,123)(40,131,110)(42,45,55)(44,98,139)(47,108,126)(50,118,113)(53,128,100)(56,138,87)(57,95,129)(60,105,116)(63,115,103)(66,125,90)(67,82,132)(69,135,77)(70,92,119)(73,102,106)(76,112,93)(79,122,80)(83,89,109)(86,99,96)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139), (2,97,43)(3,54,85)(4,11,127)(5,107,30)(6,64,72)(7,21,114)(8,117,17)(9,74,59)(10,31,101)(12,84,46)(13,41,88)(14,137,130)(15,94,33)(16,51,75)(18,104,20)(19,61,62)(22,71,49)(23,28,91)(24,124,133)(25,81,36)(26,38,78)(27,134,120)(29,48,65)(32,58,52)(34,111,136)(35,68,39)(37,121,123)(40,131,110)(42,45,55)(44,98,139)(47,108,126)(50,118,113)(53,128,100)(56,138,87)(57,95,129)(60,105,116)(63,115,103)(66,125,90)(67,82,132)(69,135,77)(70,92,119)(73,102,106)(76,112,93)(79,122,80)(83,89,109)(86,99,96) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139)], [(2,97,43),(3,54,85),(4,11,127),(5,107,30),(6,64,72),(7,21,114),(8,117,17),(9,74,59),(10,31,101),(12,84,46),(13,41,88),(14,137,130),(15,94,33),(16,51,75),(18,104,20),(19,61,62),(22,71,49),(23,28,91),(24,124,133),(25,81,36),(26,38,78),(27,134,120),(29,48,65),(32,58,52),(34,111,136),(35,68,39),(37,121,123),(40,131,110),(42,45,55),(44,98,139),(47,108,126),(50,118,113),(53,128,100),(56,138,87),(57,95,129),(60,105,116),(63,115,103),(66,125,90),(67,82,132),(69,135,77),(70,92,119),(73,102,106),(76,112,93),(79,122,80),(83,89,109),(86,99,96)]])
49 conjugacy classes
| class | 1 | 3A | 3B | 139A | ··· | 139AT |
| order | 1 | 3 | 3 | 139 | ··· | 139 |
| size | 1 | 139 | 139 | 3 | ··· | 3 |
49 irreducible representations
| dim | 1 | 1 | 3 |
| type | + | ||
| image | C1 | C3 | C139⋊C3 |
| kernel | C139⋊C3 | C139 | C1 |
| # reps | 1 | 2 | 46 |
Matrix representation of C139⋊C3 ►in GL3(𝔽1669) generated by
| 644 | 1 | 0 |
| 271 | 0 | 1 |
| 1 | 0 | 0 |
| 1 | 1550 | 534 |
| 0 | 831 | 45 |
| 0 | 1029 | 837 |
G:=sub<GL(3,GF(1669))| [644,271,1,1,0,0,0,1,0],[1,0,0,1550,831,1029,534,45,837] >;
C139⋊C3 in GAP, Magma, Sage, TeX
C_{139}\rtimes C_3 % in TeX
G:=Group("C139:C3"); // GroupNames label
G:=SmallGroup(417,1);
// by ID
G=gap.SmallGroup(417,1);
# by ID
G:=PCGroup([2,-3,-139,1153]);
// Polycyclic
G:=Group<a,b|a^139=b^3=1,b*a*b^-1=a^42>;
// generators/relations
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