direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C19, C4.C38, C76.3C2, C38.7C22, C2.2(C2×C38), SmallGroup(152,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C19
G = < a,b,c | a19=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C19
►Regular action on 152 points
Generators in S152
(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)(134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152)
(1 93 38 59)(2 94 20 60)(3 95 21 61)(4 77 22 62)(5 78 23 63)(6 79 24 64)(7 80 25 65)(8 81 26 66)(9 82 27 67)(10 83 28 68)(11 84 29 69)(12 85 30 70)(13 86 31 71)(14 87 32 72)(15 88 33 73)(16 89 34 74)(17 90 35 75)(18 91 36 76)(19 92 37 58)(39 126 142 113)(40 127 143 114)(41 128 144 96)(42 129 145 97)(43 130 146 98)(44 131 147 99)(45 132 148 100)(46 133 149 101)(47 115 150 102)(48 116 151 103)(49 117 152 104)(50 118 134 105)(51 119 135 106)(52 120 136 107)(53 121 137 108)(54 122 138 109)(55 123 139 110)(56 124 140 111)(57 125 141 112)
(1 111 38 124)(2 112 20 125)(3 113 21 126)(4 114 22 127)(5 96 23 128)(6 97 24 129)(7 98 25 130)(8 99 26 131)(9 100 27 132)(10 101 28 133)(11 102 29 115)(12 103 30 116)(13 104 31 117)(14 105 32 118)(15 106 33 119)(16 107 34 120)(17 108 35 121)(18 109 36 122)(19 110 37 123)(39 95 142 61)(40 77 143 62)(41 78 144 63)(42 79 145 64)(43 80 146 65)(44 81 147 66)(45 82 148 67)(46 83 149 68)(47 84 150 69)(48 85 151 70)(49 86 152 71)(50 87 134 72)(51 88 135 73)(52 89 136 74)(53 90 137 75)(54 91 138 76)(55 92 139 58)(56 93 140 59)(57 94 141 60)
G:=sub<Sym(152)| (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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,93,38,59)(2,94,20,60)(3,95,21,61)(4,77,22,62)(5,78,23,63)(6,79,24,64)(7,80,25,65)(8,81,26,66)(9,82,27,67)(10,83,28,68)(11,84,29,69)(12,85,30,70)(13,86,31,71)(14,87,32,72)(15,88,33,73)(16,89,34,74)(17,90,35,75)(18,91,36,76)(19,92,37,58)(39,126,142,113)(40,127,143,114)(41,128,144,96)(42,129,145,97)(43,130,146,98)(44,131,147,99)(45,132,148,100)(46,133,149,101)(47,115,150,102)(48,116,151,103)(49,117,152,104)(50,118,134,105)(51,119,135,106)(52,120,136,107)(53,121,137,108)(54,122,138,109)(55,123,139,110)(56,124,140,111)(57,125,141,112), (1,111,38,124)(2,112,20,125)(3,113,21,126)(4,114,22,127)(5,96,23,128)(6,97,24,129)(7,98,25,130)(8,99,26,131)(9,100,27,132)(10,101,28,133)(11,102,29,115)(12,103,30,116)(13,104,31,117)(14,105,32,118)(15,106,33,119)(16,107,34,120)(17,108,35,121)(18,109,36,122)(19,110,37,123)(39,95,142,61)(40,77,143,62)(41,78,144,63)(42,79,145,64)(43,80,146,65)(44,81,147,66)(45,82,148,67)(46,83,149,68)(47,84,150,69)(48,85,151,70)(49,86,152,71)(50,87,134,72)(51,88,135,73)(52,89,136,74)(53,90,137,75)(54,91,138,76)(55,92,139,58)(56,93,140,59)(57,94,141,60)>;
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)(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152), (1,93,38,59)(2,94,20,60)(3,95,21,61)(4,77,22,62)(5,78,23,63)(6,79,24,64)(7,80,25,65)(8,81,26,66)(9,82,27,67)(10,83,28,68)(11,84,29,69)(12,85,30,70)(13,86,31,71)(14,87,32,72)(15,88,33,73)(16,89,34,74)(17,90,35,75)(18,91,36,76)(19,92,37,58)(39,126,142,113)(40,127,143,114)(41,128,144,96)(42,129,145,97)(43,130,146,98)(44,131,147,99)(45,132,148,100)(46,133,149,101)(47,115,150,102)(48,116,151,103)(49,117,152,104)(50,118,134,105)(51,119,135,106)(52,120,136,107)(53,121,137,108)(54,122,138,109)(55,123,139,110)(56,124,140,111)(57,125,141,112), (1,111,38,124)(2,112,20,125)(3,113,21,126)(4,114,22,127)(5,96,23,128)(6,97,24,129)(7,98,25,130)(8,99,26,131)(9,100,27,132)(10,101,28,133)(11,102,29,115)(12,103,30,116)(13,104,31,117)(14,105,32,118)(15,106,33,119)(16,107,34,120)(17,108,35,121)(18,109,36,122)(19,110,37,123)(39,95,142,61)(40,77,143,62)(41,78,144,63)(42,79,145,64)(43,80,146,65)(44,81,147,66)(45,82,148,67)(46,83,149,68)(47,84,150,69)(48,85,151,70)(49,86,152,71)(50,87,134,72)(51,88,135,73)(52,89,136,74)(53,90,137,75)(54,91,138,76)(55,92,139,58)(56,93,140,59)(57,94,141,60) );
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),(134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152)], [(1,93,38,59),(2,94,20,60),(3,95,21,61),(4,77,22,62),(5,78,23,63),(6,79,24,64),(7,80,25,65),(8,81,26,66),(9,82,27,67),(10,83,28,68),(11,84,29,69),(12,85,30,70),(13,86,31,71),(14,87,32,72),(15,88,33,73),(16,89,34,74),(17,90,35,75),(18,91,36,76),(19,92,37,58),(39,126,142,113),(40,127,143,114),(41,128,144,96),(42,129,145,97),(43,130,146,98),(44,131,147,99),(45,132,148,100),(46,133,149,101),(47,115,150,102),(48,116,151,103),(49,117,152,104),(50,118,134,105),(51,119,135,106),(52,120,136,107),(53,121,137,108),(54,122,138,109),(55,123,139,110),(56,124,140,111),(57,125,141,112)], [(1,111,38,124),(2,112,20,125),(3,113,21,126),(4,114,22,127),(5,96,23,128),(6,97,24,129),(7,98,25,130),(8,99,26,131),(9,100,27,132),(10,101,28,133),(11,102,29,115),(12,103,30,116),(13,104,31,117),(14,105,32,118),(15,106,33,119),(16,107,34,120),(17,108,35,121),(18,109,36,122),(19,110,37,123),(39,95,142,61),(40,77,143,62),(41,78,144,63),(42,79,145,64),(43,80,146,65),(44,81,147,66),(45,82,148,67),(46,83,149,68),(47,84,150,69),(48,85,151,70),(49,86,152,71),(50,87,134,72),(51,88,135,73),(52,89,136,74),(53,90,137,75),(54,91,138,76),(55,92,139,58),(56,93,140,59),(57,94,141,60)]])
Q8×C19 is a maximal subgroup of
Q8⋊D19 C19⋊Q16 D76⋊C2 C38.A4
95 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 19A | ··· | 19R | 38A | ··· | 38R | 76A | ··· | 76BB |
| order | 1 | 2 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C19 | C38 | Q8 | Q8×C19 |
| kernel | Q8×C19 | C76 | Q8 | C4 | C19 | C1 |
| # reps | 1 | 3 | 18 | 54 | 1 | 18 |
Matrix representation of Q8×C19 ►in GL2(𝔽229) generated by
| 17 | 0 |
| 0 | 17 |
| 57 | 227 |
| 22 | 172 |
| 81 | 9 |
| 136 | 148 |
G:=sub<GL(2,GF(229))| [17,0,0,17],[57,22,227,172],[81,136,9,148] >;
Q8×C19 in GAP, Magma, Sage, TeX
Q_8\times C_{19} % in TeX
G:=Group("Q8xC19"); // GroupNames label
G:=SmallGroup(152,10);
// by ID
G=gap.SmallGroup(152,10);
# by ID
G:=PCGroup([4,-2,-2,-19,-2,304,625,309]);
// Polycyclic
G:=Group<a,b,c|a^19=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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