direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4.Q16, C12.29Q16, C4⋊C8.7C6, Q8⋊3(C3×Q8), (C3×Q8)⋊9Q8, C4⋊Q8.6C6, C2.7(C6×Q16), C4.7(C3×Q16), C4.15(C6×Q8), C2.D8.4C6, (C4×Q8).13C6, C6.54(C2×Q16), (C2×C12).333D4, C42.23(C2×C6), (Q8×C12).20C2, Q8⋊C4.3C6, C12.121(C2×Q8), C22.98(C6×D4), C6.96(C22⋊Q8), C12.314(C4○D4), C6.140(C8⋊C22), (C2×C12).933C23, (C2×C24).188C22, (C4×C12).265C22, (C6×Q8).264C22, (C2×C8).7(C2×C6), (C3×C4⋊C8).17C2, C4⋊C4.14(C2×C6), C4.26(C3×C4○D4), (C3×C4⋊Q8).21C2, (C2×C6).654(C2×D4), (C2×C4).134(C3×D4), C2.15(C3×C8⋊C22), (C2×Q8).63(C2×C6), (C3×C2.D8).13C2, C2.15(C3×C22⋊Q8), (C3×Q8⋊C4).8C2, (C3×C4⋊C4).236C22, (C2×C4).108(C22×C6), SmallGroup(192,910)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.Q16
G = < a,b,c,d | a3=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b2c-1 >
Subgroups: 154 in 96 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C2.D8, C4×Q8, C4⋊Q8, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×Q8, C6×Q8, C4.Q16, C3×Q8⋊C4, C3×C4⋊C8, C3×C2.D8, Q8×C12, C3×C4⋊Q8, C3×C4.Q16
Quotients: C1, C2, C3, C22, C6, D4, Q8, C23, C2×C6, Q16, C2×D4, C2×Q8, C4○D4, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C2×Q16, C8⋊C22, C3×Q16, C6×D4, C6×Q8, C3×C4○D4, C4.Q16, C3×C22⋊Q8, C6×Q16, C3×C8⋊C22, C3×C4.Q16
►Regular action on 192 points
Generators in S192
(1 51 163)(2 52 164)(3 53 165)(4 54 166)(5 55 167)(6 56 168)(7 49 161)(8 50 162)(9 74 139)(10 75 140)(11 76 141)(12 77 142)(13 78 143)(14 79 144)(15 80 137)(16 73 138)(17 25 99)(18 26 100)(19 27 101)(20 28 102)(21 29 103)(22 30 104)(23 31 97)(24 32 98)(33 48 105)(34 41 106)(35 42 107)(36 43 108)(37 44 109)(38 45 110)(39 46 111)(40 47 112)(57 169 177)(58 170 178)(59 171 179)(60 172 180)(61 173 181)(62 174 182)(63 175 183)(64 176 184)(65 128 185)(66 121 186)(67 122 187)(68 123 188)(69 124 189)(70 125 190)(71 126 191)(72 127 192)(81 146 154)(82 147 155)(83 148 156)(84 149 157)(85 150 158)(86 151 159)(87 152 160)(88 145 153)(89 115 132)(90 116 133)(91 117 134)(92 118 135)(93 119 136)(94 120 129)(95 113 130)(96 114 131)
(1 129 101 71)(2 72 102 130)(3 131 103 65)(4 66 104 132)(5 133 97 67)(6 68 98 134)(7 135 99 69)(8 70 100 136)(9 39 179 159)(10 160 180 40)(11 33 181 153)(12 154 182 34)(13 35 183 155)(14 156 184 36)(15 37 177 157)(16 158 178 38)(17 124 49 92)(18 93 50 125)(19 126 51 94)(20 95 52 127)(21 128 53 96)(22 89 54 121)(23 122 55 90)(24 91 56 123)(25 189 161 118)(26 119 162 190)(27 191 163 120)(28 113 164 192)(29 185 165 114)(30 115 166 186)(31 187 167 116)(32 117 168 188)(41 77 81 62)(42 63 82 78)(43 79 83 64)(44 57 84 80)(45 73 85 58)(46 59 86 74)(47 75 87 60)(48 61 88 76)(105 173 145 141)(106 142 146 174)(107 175 147 143)(108 144 148 176)(109 169 149 137)(110 138 150 170)(111 171 151 139)(112 140 152 172)
(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)
(1 151 5 147)(2 110 6 106)(3 149 7 145)(4 108 8 112)(9 90 13 94)(10 121 14 125)(11 96 15 92)(12 127 16 123)(17 33 21 37)(18 160 22 156)(19 39 23 35)(20 158 24 154)(25 48 29 44)(26 87 30 83)(27 46 31 42)(28 85 32 81)(34 52 38 56)(36 50 40 54)(41 164 45 168)(43 162 47 166)(49 153 53 157)(51 159 55 155)(57 189 61 185)(58 117 62 113)(59 187 63 191)(60 115 64 119)(65 169 69 173)(66 144 70 140)(67 175 71 171)(68 142 72 138)(73 188 77 192)(74 116 78 120)(75 186 79 190)(76 114 80 118)(82 163 86 167)(84 161 88 165)(89 184 93 180)(91 182 95 178)(97 107 101 111)(98 146 102 150)(99 105 103 109)(100 152 104 148)(122 183 126 179)(124 181 128 177)(129 139 133 143)(130 170 134 174)(131 137 135 141)(132 176 136 172)
G:=sub<Sym(192)| (1,51,163)(2,52,164)(3,53,165)(4,54,166)(5,55,167)(6,56,168)(7,49,161)(8,50,162)(9,74,139)(10,75,140)(11,76,141)(12,77,142)(13,78,143)(14,79,144)(15,80,137)(16,73,138)(17,25,99)(18,26,100)(19,27,101)(20,28,102)(21,29,103)(22,30,104)(23,31,97)(24,32,98)(33,48,105)(34,41,106)(35,42,107)(36,43,108)(37,44,109)(38,45,110)(39,46,111)(40,47,112)(57,169,177)(58,170,178)(59,171,179)(60,172,180)(61,173,181)(62,174,182)(63,175,183)(64,176,184)(65,128,185)(66,121,186)(67,122,187)(68,123,188)(69,124,189)(70,125,190)(71,126,191)(72,127,192)(81,146,154)(82,147,155)(83,148,156)(84,149,157)(85,150,158)(86,151,159)(87,152,160)(88,145,153)(89,115,132)(90,116,133)(91,117,134)(92,118,135)(93,119,136)(94,120,129)(95,113,130)(96,114,131), (1,129,101,71)(2,72,102,130)(3,131,103,65)(4,66,104,132)(5,133,97,67)(6,68,98,134)(7,135,99,69)(8,70,100,136)(9,39,179,159)(10,160,180,40)(11,33,181,153)(12,154,182,34)(13,35,183,155)(14,156,184,36)(15,37,177,157)(16,158,178,38)(17,124,49,92)(18,93,50,125)(19,126,51,94)(20,95,52,127)(21,128,53,96)(22,89,54,121)(23,122,55,90)(24,91,56,123)(25,189,161,118)(26,119,162,190)(27,191,163,120)(28,113,164,192)(29,185,165,114)(30,115,166,186)(31,187,167,116)(32,117,168,188)(41,77,81,62)(42,63,82,78)(43,79,83,64)(44,57,84,80)(45,73,85,58)(46,59,86,74)(47,75,87,60)(48,61,88,76)(105,173,145,141)(106,142,146,174)(107,175,147,143)(108,144,148,176)(109,169,149,137)(110,138,150,170)(111,171,151,139)(112,140,152,172), (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), (1,151,5,147)(2,110,6,106)(3,149,7,145)(4,108,8,112)(9,90,13,94)(10,121,14,125)(11,96,15,92)(12,127,16,123)(17,33,21,37)(18,160,22,156)(19,39,23,35)(20,158,24,154)(25,48,29,44)(26,87,30,83)(27,46,31,42)(28,85,32,81)(34,52,38,56)(36,50,40,54)(41,164,45,168)(43,162,47,166)(49,153,53,157)(51,159,55,155)(57,189,61,185)(58,117,62,113)(59,187,63,191)(60,115,64,119)(65,169,69,173)(66,144,70,140)(67,175,71,171)(68,142,72,138)(73,188,77,192)(74,116,78,120)(75,186,79,190)(76,114,80,118)(82,163,86,167)(84,161,88,165)(89,184,93,180)(91,182,95,178)(97,107,101,111)(98,146,102,150)(99,105,103,109)(100,152,104,148)(122,183,126,179)(124,181,128,177)(129,139,133,143)(130,170,134,174)(131,137,135,141)(132,176,136,172)>;
G:=Group( (1,51,163)(2,52,164)(3,53,165)(4,54,166)(5,55,167)(6,56,168)(7,49,161)(8,50,162)(9,74,139)(10,75,140)(11,76,141)(12,77,142)(13,78,143)(14,79,144)(15,80,137)(16,73,138)(17,25,99)(18,26,100)(19,27,101)(20,28,102)(21,29,103)(22,30,104)(23,31,97)(24,32,98)(33,48,105)(34,41,106)(35,42,107)(36,43,108)(37,44,109)(38,45,110)(39,46,111)(40,47,112)(57,169,177)(58,170,178)(59,171,179)(60,172,180)(61,173,181)(62,174,182)(63,175,183)(64,176,184)(65,128,185)(66,121,186)(67,122,187)(68,123,188)(69,124,189)(70,125,190)(71,126,191)(72,127,192)(81,146,154)(82,147,155)(83,148,156)(84,149,157)(85,150,158)(86,151,159)(87,152,160)(88,145,153)(89,115,132)(90,116,133)(91,117,134)(92,118,135)(93,119,136)(94,120,129)(95,113,130)(96,114,131), (1,129,101,71)(2,72,102,130)(3,131,103,65)(4,66,104,132)(5,133,97,67)(6,68,98,134)(7,135,99,69)(8,70,100,136)(9,39,179,159)(10,160,180,40)(11,33,181,153)(12,154,182,34)(13,35,183,155)(14,156,184,36)(15,37,177,157)(16,158,178,38)(17,124,49,92)(18,93,50,125)(19,126,51,94)(20,95,52,127)(21,128,53,96)(22,89,54,121)(23,122,55,90)(24,91,56,123)(25,189,161,118)(26,119,162,190)(27,191,163,120)(28,113,164,192)(29,185,165,114)(30,115,166,186)(31,187,167,116)(32,117,168,188)(41,77,81,62)(42,63,82,78)(43,79,83,64)(44,57,84,80)(45,73,85,58)(46,59,86,74)(47,75,87,60)(48,61,88,76)(105,173,145,141)(106,142,146,174)(107,175,147,143)(108,144,148,176)(109,169,149,137)(110,138,150,170)(111,171,151,139)(112,140,152,172), (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), (1,151,5,147)(2,110,6,106)(3,149,7,145)(4,108,8,112)(9,90,13,94)(10,121,14,125)(11,96,15,92)(12,127,16,123)(17,33,21,37)(18,160,22,156)(19,39,23,35)(20,158,24,154)(25,48,29,44)(26,87,30,83)(27,46,31,42)(28,85,32,81)(34,52,38,56)(36,50,40,54)(41,164,45,168)(43,162,47,166)(49,153,53,157)(51,159,55,155)(57,189,61,185)(58,117,62,113)(59,187,63,191)(60,115,64,119)(65,169,69,173)(66,144,70,140)(67,175,71,171)(68,142,72,138)(73,188,77,192)(74,116,78,120)(75,186,79,190)(76,114,80,118)(82,163,86,167)(84,161,88,165)(89,184,93,180)(91,182,95,178)(97,107,101,111)(98,146,102,150)(99,105,103,109)(100,152,104,148)(122,183,126,179)(124,181,128,177)(129,139,133,143)(130,170,134,174)(131,137,135,141)(132,176,136,172) );
G=PermutationGroup([[(1,51,163),(2,52,164),(3,53,165),(4,54,166),(5,55,167),(6,56,168),(7,49,161),(8,50,162),(9,74,139),(10,75,140),(11,76,141),(12,77,142),(13,78,143),(14,79,144),(15,80,137),(16,73,138),(17,25,99),(18,26,100),(19,27,101),(20,28,102),(21,29,103),(22,30,104),(23,31,97),(24,32,98),(33,48,105),(34,41,106),(35,42,107),(36,43,108),(37,44,109),(38,45,110),(39,46,111),(40,47,112),(57,169,177),(58,170,178),(59,171,179),(60,172,180),(61,173,181),(62,174,182),(63,175,183),(64,176,184),(65,128,185),(66,121,186),(67,122,187),(68,123,188),(69,124,189),(70,125,190),(71,126,191),(72,127,192),(81,146,154),(82,147,155),(83,148,156),(84,149,157),(85,150,158),(86,151,159),(87,152,160),(88,145,153),(89,115,132),(90,116,133),(91,117,134),(92,118,135),(93,119,136),(94,120,129),(95,113,130),(96,114,131)], [(1,129,101,71),(2,72,102,130),(3,131,103,65),(4,66,104,132),(5,133,97,67),(6,68,98,134),(7,135,99,69),(8,70,100,136),(9,39,179,159),(10,160,180,40),(11,33,181,153),(12,154,182,34),(13,35,183,155),(14,156,184,36),(15,37,177,157),(16,158,178,38),(17,124,49,92),(18,93,50,125),(19,126,51,94),(20,95,52,127),(21,128,53,96),(22,89,54,121),(23,122,55,90),(24,91,56,123),(25,189,161,118),(26,119,162,190),(27,191,163,120),(28,113,164,192),(29,185,165,114),(30,115,166,186),(31,187,167,116),(32,117,168,188),(41,77,81,62),(42,63,82,78),(43,79,83,64),(44,57,84,80),(45,73,85,58),(46,59,86,74),(47,75,87,60),(48,61,88,76),(105,173,145,141),(106,142,146,174),(107,175,147,143),(108,144,148,176),(109,169,149,137),(110,138,150,170),(111,171,151,139),(112,140,152,172)], [(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)], [(1,151,5,147),(2,110,6,106),(3,149,7,145),(4,108,8,112),(9,90,13,94),(10,121,14,125),(11,96,15,92),(12,127,16,123),(17,33,21,37),(18,160,22,156),(19,39,23,35),(20,158,24,154),(25,48,29,44),(26,87,30,83),(27,46,31,42),(28,85,32,81),(34,52,38,56),(36,50,40,54),(41,164,45,168),(43,162,47,166),(49,153,53,157),(51,159,55,155),(57,189,61,185),(58,117,62,113),(59,187,63,191),(60,115,64,119),(65,169,69,173),(66,144,70,140),(67,175,71,171),(68,142,72,138),(73,188,77,192),(74,116,78,120),(75,186,79,190),(76,114,80,118),(82,163,86,167),(84,161,88,165),(89,184,93,180),(91,182,95,178),(97,107,101,111),(98,146,102,150),(99,105,103,109),(100,152,104,148),(122,183,126,179),(124,181,128,177),(129,139,133,143),(130,170,134,174),(131,137,135,141),(132,176,136,172)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12R | 12S | 12T | 12U | 12V | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | Q8 | Q16 | C4○D4 | C3×D4 | C3×Q8 | C3×Q16 | C3×C4○D4 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C4.Q16 | C3×Q8⋊C4 | C3×C4⋊C8 | C3×C2.D8 | Q8×C12 | C3×C4⋊Q8 | C4.Q16 | Q8⋊C4 | C4⋊C8 | C2.D8 | C4×Q8 | C4⋊Q8 | C2×C12 | C3×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 1 | 2 |
Matrix representation of C3×C4.Q16 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 21 | 19 | 0 | 0 |
| 19 | 52 | 0 | 0 |
| 0 | 0 | 16 | 57 |
| 0 | 0 | 16 | 16 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 52 | 54 |
| 0 | 0 | 54 | 21 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[21,19,0,0,19,52,0,0,0,0,16,16,0,0,57,16],[72,0,0,0,0,72,0,0,0,0,52,54,0,0,54,21] >;
C3×C4.Q16 in GAP, Magma, Sage, TeX
C_3\times C_4.Q_{16} % in TeX
G:=Group("C3xC4.Q16"); // GroupNames label
G:=SmallGroup(192,910);
// by ID
G=gap.SmallGroup(192,910);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,840,365,176,1094,520,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c^-1>;
// generators/relations