direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×C3⋊C16, C24.76C23, (C2×C6)⋊3C16, C6⋊2(C2×C16), C3⋊2(C22×C16), C24.83(C2×C4), (C2×C24).27C4, (C2×C12).12C8, C12.44(C2×C8), (C2×C8).343D6, C23.5(C3⋊C8), (C22×C6).6C8, C6.26(C22×C8), C8.62(C22×S3), (C22×C8).19S3, (C2×C8).19Dic3, C8.25(C2×Dic3), (C22×C24).26C2, (C22×C12).32C4, (C2×C24).430C22, C12.175(C22×C4), C4.29(C22×Dic3), (C22×C4).23Dic3, C4.17(C2×C3⋊C8), (C2×C4).9(C3⋊C8), C2.2(C22×C3⋊C8), (C2×C6).36(C2×C8), C22.14(C2×C3⋊C8), (C2×C12).318(C2×C4), (C2×C4).100(C2×Dic3), SmallGroup(192,655)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C22×C3⋊C16 |
Generators and relations for C22×C3⋊C16
G = < a,b,c,d | a2=b2=c3=d16=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 120 in 98 conjugacy classes, 87 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C23, C12, C12, C2×C6, C16, C2×C8, C22×C4, C24, C24, C2×C12, C22×C6, C2×C16, C22×C8, C3⋊C16, C2×C24, C22×C12, C22×C16, C2×C3⋊C16, C22×C24, C22×C3⋊C16
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C16, C2×C8, C22×C4, C3⋊C8, C2×Dic3, C22×S3, C2×C16, C22×C8, C3⋊C16, C2×C3⋊C8, C22×Dic3, C22×C16, C2×C3⋊C16, C22×C3⋊C8, C22×C3⋊C16
►Regular action on 192 points
Generators in S192
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 97)(15 98)(16 99)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 92)(34 93)(35 94)(36 95)(37 96)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 191)(58 192)(59 177)(60 178)(61 179)(62 180)(63 181)(64 182)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)(121 158)(122 159)(123 160)(124 145)(125 146)(126 147)(127 148)(128 149)(129 174)(130 175)(131 176)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)
(1 95)(2 96)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 184)(18 185)(19 186)(20 187)(21 188)(22 189)(23 190)(24 191)(25 192)(26 177)(27 178)(28 179)(29 180)(30 181)(31 182)(32 183)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 80)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(113 144)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(145 168)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 161)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)
(1 157 70)(2 71 158)(3 159 72)(4 73 160)(5 145 74)(6 75 146)(7 147 76)(8 77 148)(9 149 78)(10 79 150)(11 151 80)(12 65 152)(13 153 66)(14 67 154)(15 155 68)(16 69 156)(17 115 111)(18 112 116)(19 117 97)(20 98 118)(21 119 99)(22 100 120)(23 121 101)(24 102 122)(25 123 103)(26 104 124)(27 125 105)(28 106 126)(29 127 107)(30 108 128)(31 113 109)(32 110 114)(33 186 132)(34 133 187)(35 188 134)(36 135 189)(37 190 136)(38 137 191)(39 192 138)(40 139 177)(41 178 140)(42 141 179)(43 180 142)(44 143 181)(45 182 144)(46 129 183)(47 184 130)(48 131 185)(49 89 174)(50 175 90)(51 91 176)(52 161 92)(53 93 162)(54 163 94)(55 95 164)(56 165 96)(57 81 166)(58 167 82)(59 83 168)(60 169 84)(61 85 170)(62 171 86)(63 87 172)(64 173 88)
(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)
G:=sub<Sym(192)| (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,97)(15,98)(16,99)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,92)(34,93)(35,94)(36,95)(37,96)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149)(129,174)(130,175)(131,176)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173), (1,95)(2,96)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,80)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(113,144)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167), (1,157,70)(2,71,158)(3,159,72)(4,73,160)(5,145,74)(6,75,146)(7,147,76)(8,77,148)(9,149,78)(10,79,150)(11,151,80)(12,65,152)(13,153,66)(14,67,154)(15,155,68)(16,69,156)(17,115,111)(18,112,116)(19,117,97)(20,98,118)(21,119,99)(22,100,120)(23,121,101)(24,102,122)(25,123,103)(26,104,124)(27,125,105)(28,106,126)(29,127,107)(30,108,128)(31,113,109)(32,110,114)(33,186,132)(34,133,187)(35,188,134)(36,135,189)(37,190,136)(38,137,191)(39,192,138)(40,139,177)(41,178,140)(42,141,179)(43,180,142)(44,143,181)(45,182,144)(46,129,183)(47,184,130)(48,131,185)(49,89,174)(50,175,90)(51,91,176)(52,161,92)(53,93,162)(54,163,94)(55,95,164)(56,165,96)(57,81,166)(58,167,82)(59,83,168)(60,169,84)(61,85,170)(62,171,86)(63,87,172)(64,173,88), (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)>;
G:=Group( (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,97)(15,98)(16,99)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,92)(34,93)(35,94)(36,95)(37,96)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,191)(58,192)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(113,150)(114,151)(115,152)(116,153)(117,154)(118,155)(119,156)(120,157)(121,158)(122,159)(123,160)(124,145)(125,146)(126,147)(127,148)(128,149)(129,174)(130,175)(131,176)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173), (1,95)(2,96)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,91)(14,92)(15,93)(16,94)(17,184)(18,185)(19,186)(20,187)(21,188)(22,189)(23,190)(24,191)(25,192)(26,177)(27,178)(28,179)(29,180)(30,181)(31,182)(32,183)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,80)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(113,144)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,141)(127,142)(128,143)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167), (1,157,70)(2,71,158)(3,159,72)(4,73,160)(5,145,74)(6,75,146)(7,147,76)(8,77,148)(9,149,78)(10,79,150)(11,151,80)(12,65,152)(13,153,66)(14,67,154)(15,155,68)(16,69,156)(17,115,111)(18,112,116)(19,117,97)(20,98,118)(21,119,99)(22,100,120)(23,121,101)(24,102,122)(25,123,103)(26,104,124)(27,125,105)(28,106,126)(29,127,107)(30,108,128)(31,113,109)(32,110,114)(33,186,132)(34,133,187)(35,188,134)(36,135,189)(37,190,136)(38,137,191)(39,192,138)(40,139,177)(41,178,140)(42,141,179)(43,180,142)(44,143,181)(45,182,144)(46,129,183)(47,184,130)(48,131,185)(49,89,174)(50,175,90)(51,91,176)(52,161,92)(53,93,162)(54,163,94)(55,95,164)(56,165,96)(57,81,166)(58,167,82)(59,83,168)(60,169,84)(61,85,170)(62,171,86)(63,87,172)(64,173,88), (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) );
G=PermutationGroup([[(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,97),(15,98),(16,99),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,92),(34,93),(35,94),(36,95),(37,96),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,183),(50,184),(51,185),(52,186),(53,187),(54,188),(55,189),(56,190),(57,191),(58,192),(59,177),(60,178),(61,179),(62,180),(63,181),(64,182),(113,150),(114,151),(115,152),(116,153),(117,154),(118,155),(119,156),(120,157),(121,158),(122,159),(123,160),(124,145),(125,146),(126,147),(127,148),(128,149),(129,174),(130,175),(131,176),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173)], [(1,95),(2,96),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,91),(14,92),(15,93),(16,94),(17,184),(18,185),(19,186),(20,187),(21,188),(22,189),(23,190),(24,191),(25,192),(26,177),(27,178),(28,179),(29,180),(30,181),(31,182),(32,183),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,80),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(113,144),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135),(121,136),(122,137),(123,138),(124,139),(125,140),(126,141),(127,142),(128,143),(145,168),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,161),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167)], [(1,157,70),(2,71,158),(3,159,72),(4,73,160),(5,145,74),(6,75,146),(7,147,76),(8,77,148),(9,149,78),(10,79,150),(11,151,80),(12,65,152),(13,153,66),(14,67,154),(15,155,68),(16,69,156),(17,115,111),(18,112,116),(19,117,97),(20,98,118),(21,119,99),(22,100,120),(23,121,101),(24,102,122),(25,123,103),(26,104,124),(27,125,105),(28,106,126),(29,127,107),(30,108,128),(31,113,109),(32,110,114),(33,186,132),(34,133,187),(35,188,134),(36,135,189),(37,190,136),(38,137,191),(39,192,138),(40,139,177),(41,178,140),(42,141,179),(43,180,142),(44,143,181),(45,182,144),(46,129,183),(47,184,130),(48,131,185),(49,89,174),(50,175,90),(51,91,176),(52,161,92),(53,93,162),(54,163,94),(55,95,164),(56,165,96),(57,81,166),(58,167,82),(59,83,168),(60,169,84),(61,85,170),(62,171,86),(63,87,172),(64,173,88)], [(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)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 6A | ··· | 6G | 8A | ··· | 8P | 12A | ··· | 12H | 16A | ··· | 16AF | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | C3⋊C16 |
| kernel | C22×C3⋊C16 | C2×C3⋊C16 | C22×C24 | C2×C24 | C22×C12 | C2×C12 | C22×C6 | C2×C6 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 12 | 4 | 32 | 1 | 3 | 3 | 1 | 6 | 2 | 16 |
Matrix representation of C22×C3⋊C16 ►in GL4(𝔽97) generated by
| 1 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 96 |
| 96 | 0 | 0 | 0 |
| 0 | 96 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 96 |
| 0 | 0 | 1 | 96 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 8 | 89 |
G:=sub<GL(4,GF(97))| [1,0,0,0,0,96,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,96,96],[1,0,0,0,0,1,0,0,0,0,8,8,0,0,0,89] >;
C22×C3⋊C16 in GAP, Magma, Sage, TeX
C_2^2\times C_3\rtimes C_{16} % in TeX
G:=Group("C2^2xC3:C16"); // GroupNames label
G:=SmallGroup(192,655);
// by ID
G=gap.SmallGroup(192,655);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^3=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations