direct product, metacyclic, nilpotent (class 5), monomial, 2-elementary
Aliases: C3×Q64, C32.C6, Q32.C6, C96.2C2, C24.66D4, C6.17D16, C12.41D8, C48.21C22, C8.7(C3×D4), C4.3(C3×D8), C16.4(C2×C6), C2.5(C3×D16), (C3×Q32).2C2, SmallGroup(192,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q64
G = < a,b,c | a3=b32=1, c2=b16, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Q64
►Regular action on 192 points
Generators in S192
(1 96 177)(2 65 178)(3 66 179)(4 67 180)(5 68 181)(6 69 182)(7 70 183)(8 71 184)(9 72 185)(10 73 186)(11 74 187)(12 75 188)(13 76 189)(14 77 190)(15 78 191)(16 79 192)(17 80 161)(18 81 162)(19 82 163)(20 83 164)(21 84 165)(22 85 166)(23 86 167)(24 87 168)(25 88 169)(26 89 170)(27 90 171)(28 91 172)(29 92 173)(30 93 174)(31 94 175)(32 95 176)(33 148 114)(34 149 115)(35 150 116)(36 151 117)(37 152 118)(38 153 119)(39 154 120)(40 155 121)(41 156 122)(42 157 123)(43 158 124)(44 159 125)(45 160 126)(46 129 127)(47 130 128)(48 131 97)(49 132 98)(50 133 99)(51 134 100)(52 135 101)(53 136 102)(54 137 103)(55 138 104)(56 139 105)(57 140 106)(58 141 107)(59 142 108)(60 143 109)(61 144 110)(62 145 111)(63 146 112)(64 147 113)
(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)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192)
(1 157 17 141)(2 156 18 140)(3 155 19 139)(4 154 20 138)(5 153 21 137)(6 152 22 136)(7 151 23 135)(8 150 24 134)(9 149 25 133)(10 148 26 132)(11 147 27 131)(12 146 28 130)(13 145 29 129)(14 144 30 160)(15 143 31 159)(16 142 32 158)(33 170 49 186)(34 169 50 185)(35 168 51 184)(36 167 52 183)(37 166 53 182)(38 165 54 181)(39 164 55 180)(40 163 56 179)(41 162 57 178)(42 161 58 177)(43 192 59 176)(44 191 60 175)(45 190 61 174)(46 189 62 173)(47 188 63 172)(48 187 64 171)(65 122 81 106)(66 121 82 105)(67 120 83 104)(68 119 84 103)(69 118 85 102)(70 117 86 101)(71 116 87 100)(72 115 88 99)(73 114 89 98)(74 113 90 97)(75 112 91 128)(76 111 92 127)(77 110 93 126)(78 109 94 125)(79 108 95 124)(80 107 96 123)
G:=sub<Sym(192)| (1,96,177)(2,65,178)(3,66,179)(4,67,180)(5,68,181)(6,69,182)(7,70,183)(8,71,184)(9,72,185)(10,73,186)(11,74,187)(12,75,188)(13,76,189)(14,77,190)(15,78,191)(16,79,192)(17,80,161)(18,81,162)(19,82,163)(20,83,164)(21,84,165)(22,85,166)(23,86,167)(24,87,168)(25,88,169)(26,89,170)(27,90,171)(28,91,172)(29,92,173)(30,93,174)(31,94,175)(32,95,176)(33,148,114)(34,149,115)(35,150,116)(36,151,117)(37,152,118)(38,153,119)(39,154,120)(40,155,121)(41,156,122)(42,157,123)(43,158,124)(44,159,125)(45,160,126)(46,129,127)(47,130,128)(48,131,97)(49,132,98)(50,133,99)(51,134,100)(52,135,101)(53,136,102)(54,137,103)(55,138,104)(56,139,105)(57,140,106)(58,141,107)(59,142,108)(60,143,109)(61,144,110)(62,145,111)(63,146,112)(64,147,113), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,157,17,141)(2,156,18,140)(3,155,19,139)(4,154,20,138)(5,153,21,137)(6,152,22,136)(7,151,23,135)(8,150,24,134)(9,149,25,133)(10,148,26,132)(11,147,27,131)(12,146,28,130)(13,145,29,129)(14,144,30,160)(15,143,31,159)(16,142,32,158)(33,170,49,186)(34,169,50,185)(35,168,51,184)(36,167,52,183)(37,166,53,182)(38,165,54,181)(39,164,55,180)(40,163,56,179)(41,162,57,178)(42,161,58,177)(43,192,59,176)(44,191,60,175)(45,190,61,174)(46,189,62,173)(47,188,63,172)(48,187,64,171)(65,122,81,106)(66,121,82,105)(67,120,83,104)(68,119,84,103)(69,118,85,102)(70,117,86,101)(71,116,87,100)(72,115,88,99)(73,114,89,98)(74,113,90,97)(75,112,91,128)(76,111,92,127)(77,110,93,126)(78,109,94,125)(79,108,95,124)(80,107,96,123)>;
G:=Group( (1,96,177)(2,65,178)(3,66,179)(4,67,180)(5,68,181)(6,69,182)(7,70,183)(8,71,184)(9,72,185)(10,73,186)(11,74,187)(12,75,188)(13,76,189)(14,77,190)(15,78,191)(16,79,192)(17,80,161)(18,81,162)(19,82,163)(20,83,164)(21,84,165)(22,85,166)(23,86,167)(24,87,168)(25,88,169)(26,89,170)(27,90,171)(28,91,172)(29,92,173)(30,93,174)(31,94,175)(32,95,176)(33,148,114)(34,149,115)(35,150,116)(36,151,117)(37,152,118)(38,153,119)(39,154,120)(40,155,121)(41,156,122)(42,157,123)(43,158,124)(44,159,125)(45,160,126)(46,129,127)(47,130,128)(48,131,97)(49,132,98)(50,133,99)(51,134,100)(52,135,101)(53,136,102)(54,137,103)(55,138,104)(56,139,105)(57,140,106)(58,141,107)(59,142,108)(60,143,109)(61,144,110)(62,145,111)(63,146,112)(64,147,113), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192), (1,157,17,141)(2,156,18,140)(3,155,19,139)(4,154,20,138)(5,153,21,137)(6,152,22,136)(7,151,23,135)(8,150,24,134)(9,149,25,133)(10,148,26,132)(11,147,27,131)(12,146,28,130)(13,145,29,129)(14,144,30,160)(15,143,31,159)(16,142,32,158)(33,170,49,186)(34,169,50,185)(35,168,51,184)(36,167,52,183)(37,166,53,182)(38,165,54,181)(39,164,55,180)(40,163,56,179)(41,162,57,178)(42,161,58,177)(43,192,59,176)(44,191,60,175)(45,190,61,174)(46,189,62,173)(47,188,63,172)(48,187,64,171)(65,122,81,106)(66,121,82,105)(67,120,83,104)(68,119,84,103)(69,118,85,102)(70,117,86,101)(71,116,87,100)(72,115,88,99)(73,114,89,98)(74,113,90,97)(75,112,91,128)(76,111,92,127)(77,110,93,126)(78,109,94,125)(79,108,95,124)(80,107,96,123) );
G=PermutationGroup([[(1,96,177),(2,65,178),(3,66,179),(4,67,180),(5,68,181),(6,69,182),(7,70,183),(8,71,184),(9,72,185),(10,73,186),(11,74,187),(12,75,188),(13,76,189),(14,77,190),(15,78,191),(16,79,192),(17,80,161),(18,81,162),(19,82,163),(20,83,164),(21,84,165),(22,85,166),(23,86,167),(24,87,168),(25,88,169),(26,89,170),(27,90,171),(28,91,172),(29,92,173),(30,93,174),(31,94,175),(32,95,176),(33,148,114),(34,149,115),(35,150,116),(36,151,117),(37,152,118),(38,153,119),(39,154,120),(40,155,121),(41,156,122),(42,157,123),(43,158,124),(44,159,125),(45,160,126),(46,129,127),(47,130,128),(48,131,97),(49,132,98),(50,133,99),(51,134,100),(52,135,101),(53,136,102),(54,137,103),(55,138,104),(56,139,105),(57,140,106),(58,141,107),(59,142,108),(60,143,109),(61,144,110),(62,145,111),(63,146,112),(64,147,113)], [(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),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192)], [(1,157,17,141),(2,156,18,140),(3,155,19,139),(4,154,20,138),(5,153,21,137),(6,152,22,136),(7,151,23,135),(8,150,24,134),(9,149,25,133),(10,148,26,132),(11,147,27,131),(12,146,28,130),(13,145,29,129),(14,144,30,160),(15,143,31,159),(16,142,32,158),(33,170,49,186),(34,169,50,185),(35,168,51,184),(36,167,52,183),(37,166,53,182),(38,165,54,181),(39,164,55,180),(40,163,56,179),(41,162,57,178),(42,161,58,177),(43,192,59,176),(44,191,60,175),(45,190,61,174),(46,189,62,173),(47,188,63,172),(48,187,64,171),(65,122,81,106),(66,121,82,105),(67,120,83,104),(68,119,84,103),(69,118,85,102),(70,117,86,101),(71,116,87,100),(72,115,88,99),(73,114,89,98),(74,113,90,97),(75,112,91,128),(76,111,92,127),(77,110,93,126),(78,109,94,125),(79,108,95,124),(80,107,96,123)]])
57 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 32A | ··· | 32H | 48A | ··· | 48H | 96A | ··· | 96P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 32 | ··· | 32 | 48 | ··· | 48 | 96 | ··· | 96 |
| size | 1 | 1 | 1 | 1 | 2 | 16 | 16 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D8 | C3×D4 | D16 | C3×D8 | Q64 | C3×D16 | C3×Q64 |
| kernel | C3×Q64 | C96 | C3×Q32 | Q64 | C32 | Q32 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×Q64 ►in GL2(𝔽31) generated by
| 25 | 0 |
| 0 | 25 |
| 0 | 30 |
| 1 | 21 |
| 8 | 18 |
| 5 | 23 |
G:=sub<GL(2,GF(31))| [25,0,0,25],[0,1,30,21],[8,5,18,23] >;
C3×Q64 in GAP, Magma, Sage, TeX
C_3\times Q_{64} % in TeX
G:=Group("C3xQ64"); // GroupNames label
G:=SmallGroup(192,179);
// by ID
G=gap.SmallGroup(192,179);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,672,197,680,1011,514,192,2524,1271,242,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^32=1,c^2=b^16,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export