direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C3⋊Q16, C12⋊8Q16, C42.212D6, C3⋊4(C4×Q16), C6.75(C4×D4), C4⋊C4.255D6, (C4×Q8).13S3, Q8.10(C4×S3), (Q8×C12).7C2, C6.35(C2×Q16), C6.94(C4○D8), (C2×C12).259D4, (C2×Q8).186D6, C4.42(C4○D12), C12.62(C4○D4), C12.27(C22×C4), Dic6.16(C2×C4), (C4×Dic6).14C2, C6.Q16.18C2, (C4×C12).100C22, (C2×C12).349C23, Q8⋊2Dic3.17C2, C2.6(Q8.13D6), C6.SD16.17C2, (C6×Q8).197C22, C4⋊Dic3.332C22, (C2×Dic6).266C22, (C4×C3⋊C8).9C2, C3⋊C8.9(C2×C4), C4.27(S3×C2×C4), C2.21(C4×C3⋊D4), C2.3(C2×C3⋊Q16), (C2×C6).480(C2×D4), (C3×Q8).12(C2×C4), (C2×C3⋊C8).247C22, (C2×C3⋊Q16).10C2, C22.81(C2×C3⋊D4), (C2×C4).104(C3⋊D4), (C3×C4⋊C4).286C22, (C2×C4).449(C22×S3), SmallGroup(192,588)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3⋊Q16
G = < a,b,c,d | a4=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 232 in 110 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C4×C8, Q8⋊C4, C2.D8, C4×Q8, C4×Q8, C2×Q16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3⋊Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4×Q16, C4×C3⋊C8, C6.Q16, C6.SD16, Q8⋊2Dic3, C4×Dic6, C2×C3⋊Q16, Q8×C12, C4×C3⋊Q16
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, Q16, C22×C4, C2×D4, C4○D4, C4×S3, C3⋊D4, C22×S3, C4×D4, C2×Q16, C4○D8, C3⋊Q16, S3×C2×C4, C4○D12, C2×C3⋊D4, C4×Q16, C4×C3⋊D4, C2×C3⋊Q16, Q8.13D6, C4×C3⋊Q16
►Regular action on 192 points
Generators in S192
(1 107 53 15)(2 108 54 16)(3 109 55 9)(4 110 56 10)(5 111 49 11)(6 112 50 12)(7 105 51 13)(8 106 52 14)(17 135 95 173)(18 136 96 174)(19 129 89 175)(20 130 90 176)(21 131 91 169)(22 132 92 170)(23 133 93 171)(24 134 94 172)(25 147 87 126)(26 148 88 127)(27 149 81 128)(28 150 82 121)(29 151 83 122)(30 152 84 123)(31 145 85 124)(32 146 86 125)(33 69 160 178)(34 70 153 179)(35 71 154 180)(36 72 155 181)(37 65 156 182)(38 66 157 183)(39 67 158 184)(40 68 159 177)(41 141 78 120)(42 142 79 113)(43 143 80 114)(44 144 73 115)(45 137 74 116)(46 138 75 117)(47 139 76 118)(48 140 77 119)(57 104 164 185)(58 97 165 186)(59 98 166 187)(60 99 167 188)(61 100 168 189)(62 101 161 190)(63 102 162 191)(64 103 163 192)
(1 65 187)(2 188 66)(3 67 189)(4 190 68)(5 69 191)(6 192 70)(7 71 185)(8 186 72)(9 39 168)(10 161 40)(11 33 162)(12 163 34)(13 35 164)(14 165 36)(15 37 166)(16 167 38)(17 118 151)(18 152 119)(19 120 145)(20 146 113)(21 114 147)(22 148 115)(23 116 149)(24 150 117)(25 169 80)(26 73 170)(27 171 74)(28 75 172)(29 173 76)(30 77 174)(31 175 78)(32 79 176)(41 85 129)(42 130 86)(43 87 131)(44 132 88)(45 81 133)(46 134 82)(47 83 135)(48 136 84)(49 178 102)(50 103 179)(51 180 104)(52 97 181)(53 182 98)(54 99 183)(55 184 100)(56 101 177)(57 105 154)(58 155 106)(59 107 156)(60 157 108)(61 109 158)(62 159 110)(63 111 160)(64 153 112)(89 141 124)(90 125 142)(91 143 126)(92 127 144)(93 137 128)(94 121 138)(95 139 122)(96 123 140)
(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 95 5 91)(2 94 6 90)(3 93 7 89)(4 92 8 96)(9 133 13 129)(10 132 14 136)(11 131 15 135)(12 130 16 134)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(25 59 29 63)(26 58 30 62)(27 57 31 61)(28 64 32 60)(33 43 37 47)(34 42 38 46)(35 41 39 45)(36 48 40 44)(65 139 69 143)(66 138 70 142)(67 137 71 141)(68 144 72 140)(73 155 77 159)(74 154 78 158)(75 153 79 157)(76 160 80 156)(81 164 85 168)(82 163 86 167)(83 162 87 166)(84 161 88 165)(97 152 101 148)(98 151 102 147)(99 150 103 146)(100 149 104 145)(105 175 109 171)(106 174 110 170)(107 173 111 169)(108 172 112 176)(113 183 117 179)(114 182 118 178)(115 181 119 177)(116 180 120 184)(121 192 125 188)(122 191 126 187)(123 190 127 186)(124 189 128 185)
G:=sub<Sym(192)| (1,107,53,15)(2,108,54,16)(3,109,55,9)(4,110,56,10)(5,111,49,11)(6,112,50,12)(7,105,51,13)(8,106,52,14)(17,135,95,173)(18,136,96,174)(19,129,89,175)(20,130,90,176)(21,131,91,169)(22,132,92,170)(23,133,93,171)(24,134,94,172)(25,147,87,126)(26,148,88,127)(27,149,81,128)(28,150,82,121)(29,151,83,122)(30,152,84,123)(31,145,85,124)(32,146,86,125)(33,69,160,178)(34,70,153,179)(35,71,154,180)(36,72,155,181)(37,65,156,182)(38,66,157,183)(39,67,158,184)(40,68,159,177)(41,141,78,120)(42,142,79,113)(43,143,80,114)(44,144,73,115)(45,137,74,116)(46,138,75,117)(47,139,76,118)(48,140,77,119)(57,104,164,185)(58,97,165,186)(59,98,166,187)(60,99,167,188)(61,100,168,189)(62,101,161,190)(63,102,162,191)(64,103,163,192), (1,65,187)(2,188,66)(3,67,189)(4,190,68)(5,69,191)(6,192,70)(7,71,185)(8,186,72)(9,39,168)(10,161,40)(11,33,162)(12,163,34)(13,35,164)(14,165,36)(15,37,166)(16,167,38)(17,118,151)(18,152,119)(19,120,145)(20,146,113)(21,114,147)(22,148,115)(23,116,149)(24,150,117)(25,169,80)(26,73,170)(27,171,74)(28,75,172)(29,173,76)(30,77,174)(31,175,78)(32,79,176)(41,85,129)(42,130,86)(43,87,131)(44,132,88)(45,81,133)(46,134,82)(47,83,135)(48,136,84)(49,178,102)(50,103,179)(51,180,104)(52,97,181)(53,182,98)(54,99,183)(55,184,100)(56,101,177)(57,105,154)(58,155,106)(59,107,156)(60,157,108)(61,109,158)(62,159,110)(63,111,160)(64,153,112)(89,141,124)(90,125,142)(91,143,126)(92,127,144)(93,137,128)(94,121,138)(95,139,122)(96,123,140), (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), (1,95,5,91)(2,94,6,90)(3,93,7,89)(4,92,8,96)(9,133,13,129)(10,132,14,136)(11,131,15,135)(12,130,16,134)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60)(33,43,37,47)(34,42,38,46)(35,41,39,45)(36,48,40,44)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156)(81,164,85,168)(82,163,86,167)(83,162,87,166)(84,161,88,165)(97,152,101,148)(98,151,102,147)(99,150,103,146)(100,149,104,145)(105,175,109,171)(106,174,110,170)(107,173,111,169)(108,172,112,176)(113,183,117,179)(114,182,118,178)(115,181,119,177)(116,180,120,184)(121,192,125,188)(122,191,126,187)(123,190,127,186)(124,189,128,185)>;
G:=Group( (1,107,53,15)(2,108,54,16)(3,109,55,9)(4,110,56,10)(5,111,49,11)(6,112,50,12)(7,105,51,13)(8,106,52,14)(17,135,95,173)(18,136,96,174)(19,129,89,175)(20,130,90,176)(21,131,91,169)(22,132,92,170)(23,133,93,171)(24,134,94,172)(25,147,87,126)(26,148,88,127)(27,149,81,128)(28,150,82,121)(29,151,83,122)(30,152,84,123)(31,145,85,124)(32,146,86,125)(33,69,160,178)(34,70,153,179)(35,71,154,180)(36,72,155,181)(37,65,156,182)(38,66,157,183)(39,67,158,184)(40,68,159,177)(41,141,78,120)(42,142,79,113)(43,143,80,114)(44,144,73,115)(45,137,74,116)(46,138,75,117)(47,139,76,118)(48,140,77,119)(57,104,164,185)(58,97,165,186)(59,98,166,187)(60,99,167,188)(61,100,168,189)(62,101,161,190)(63,102,162,191)(64,103,163,192), (1,65,187)(2,188,66)(3,67,189)(4,190,68)(5,69,191)(6,192,70)(7,71,185)(8,186,72)(9,39,168)(10,161,40)(11,33,162)(12,163,34)(13,35,164)(14,165,36)(15,37,166)(16,167,38)(17,118,151)(18,152,119)(19,120,145)(20,146,113)(21,114,147)(22,148,115)(23,116,149)(24,150,117)(25,169,80)(26,73,170)(27,171,74)(28,75,172)(29,173,76)(30,77,174)(31,175,78)(32,79,176)(41,85,129)(42,130,86)(43,87,131)(44,132,88)(45,81,133)(46,134,82)(47,83,135)(48,136,84)(49,178,102)(50,103,179)(51,180,104)(52,97,181)(53,182,98)(54,99,183)(55,184,100)(56,101,177)(57,105,154)(58,155,106)(59,107,156)(60,157,108)(61,109,158)(62,159,110)(63,111,160)(64,153,112)(89,141,124)(90,125,142)(91,143,126)(92,127,144)(93,137,128)(94,121,138)(95,139,122)(96,123,140), (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), (1,95,5,91)(2,94,6,90)(3,93,7,89)(4,92,8,96)(9,133,13,129)(10,132,14,136)(11,131,15,135)(12,130,16,134)(17,49,21,53)(18,56,22,52)(19,55,23,51)(20,54,24,50)(25,59,29,63)(26,58,30,62)(27,57,31,61)(28,64,32,60)(33,43,37,47)(34,42,38,46)(35,41,39,45)(36,48,40,44)(65,139,69,143)(66,138,70,142)(67,137,71,141)(68,144,72,140)(73,155,77,159)(74,154,78,158)(75,153,79,157)(76,160,80,156)(81,164,85,168)(82,163,86,167)(83,162,87,166)(84,161,88,165)(97,152,101,148)(98,151,102,147)(99,150,103,146)(100,149,104,145)(105,175,109,171)(106,174,110,170)(107,173,111,169)(108,172,112,176)(113,183,117,179)(114,182,118,178)(115,181,119,177)(116,180,120,184)(121,192,125,188)(122,191,126,187)(123,190,127,186)(124,189,128,185) );
G=PermutationGroup([[(1,107,53,15),(2,108,54,16),(3,109,55,9),(4,110,56,10),(5,111,49,11),(6,112,50,12),(7,105,51,13),(8,106,52,14),(17,135,95,173),(18,136,96,174),(19,129,89,175),(20,130,90,176),(21,131,91,169),(22,132,92,170),(23,133,93,171),(24,134,94,172),(25,147,87,126),(26,148,88,127),(27,149,81,128),(28,150,82,121),(29,151,83,122),(30,152,84,123),(31,145,85,124),(32,146,86,125),(33,69,160,178),(34,70,153,179),(35,71,154,180),(36,72,155,181),(37,65,156,182),(38,66,157,183),(39,67,158,184),(40,68,159,177),(41,141,78,120),(42,142,79,113),(43,143,80,114),(44,144,73,115),(45,137,74,116),(46,138,75,117),(47,139,76,118),(48,140,77,119),(57,104,164,185),(58,97,165,186),(59,98,166,187),(60,99,167,188),(61,100,168,189),(62,101,161,190),(63,102,162,191),(64,103,163,192)], [(1,65,187),(2,188,66),(3,67,189),(4,190,68),(5,69,191),(6,192,70),(7,71,185),(8,186,72),(9,39,168),(10,161,40),(11,33,162),(12,163,34),(13,35,164),(14,165,36),(15,37,166),(16,167,38),(17,118,151),(18,152,119),(19,120,145),(20,146,113),(21,114,147),(22,148,115),(23,116,149),(24,150,117),(25,169,80),(26,73,170),(27,171,74),(28,75,172),(29,173,76),(30,77,174),(31,175,78),(32,79,176),(41,85,129),(42,130,86),(43,87,131),(44,132,88),(45,81,133),(46,134,82),(47,83,135),(48,136,84),(49,178,102),(50,103,179),(51,180,104),(52,97,181),(53,182,98),(54,99,183),(55,184,100),(56,101,177),(57,105,154),(58,155,106),(59,107,156),(60,157,108),(61,109,158),(62,159,110),(63,111,160),(64,153,112),(89,141,124),(90,125,142),(91,143,126),(92,127,144),(93,137,128),(94,121,138),(95,139,122),(96,123,140)], [(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)], [(1,95,5,91),(2,94,6,90),(3,93,7,89),(4,92,8,96),(9,133,13,129),(10,132,14,136),(11,131,15,135),(12,130,16,134),(17,49,21,53),(18,56,22,52),(19,55,23,51),(20,54,24,50),(25,59,29,63),(26,58,30,62),(27,57,31,61),(28,64,32,60),(33,43,37,47),(34,42,38,46),(35,41,39,45),(36,48,40,44),(65,139,69,143),(66,138,70,142),(67,137,71,141),(68,144,72,140),(73,155,77,159),(74,154,78,158),(75,153,79,157),(76,160,80,156),(81,164,85,168),(82,163,86,167),(83,162,87,166),(84,161,88,165),(97,152,101,148),(98,151,102,147),(99,150,103,146),(100,149,104,145),(105,175,109,171),(106,174,110,170),(107,173,111,169),(108,172,112,176),(113,183,117,179),(114,182,118,178),(115,181,119,177),(116,180,120,184),(121,192,125,188),(122,191,126,187),(123,190,127,186),(124,189,128,185)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | Q16 | C4○D4 | C3⋊D4 | C4×S3 | C4○D8 | C4○D12 | C3⋊Q16 | Q8.13D6 |
| kernel | C4×C3⋊Q16 | C4×C3⋊C8 | C6.Q16 | C6.SD16 | Q8⋊2Dic3 | C4×Dic6 | C2×C3⋊Q16 | Q8×C12 | C3⋊Q16 | C4×Q8 | C2×C12 | C42 | C4⋊C4 | C2×Q8 | C12 | C12 | C2×C4 | Q8 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of C4×C3⋊Q16 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 12 | 10 |
| 0 | 0 | 22 | 61 |
| 19 | 52 | 0 | 0 |
| 52 | 54 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[16,16,0,0,57,16,0,0,0,0,12,22,0,0,10,61],[19,52,0,0,52,54,0,0,0,0,43,13,0,0,60,30] >;
C4×C3⋊Q16 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes Q_{16} % in TeX
G:=Group("C4xC3:Q16"); // GroupNames label
G:=SmallGroup(192,588);
// by ID
G=gap.SmallGroup(192,588);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,232,58,1684,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=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=c^-1>;
// generators/relations