direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C16⋊3C4, C48⋊7C4, C16⋊3C12, C6.7Q32, C6.14D16, C24.15Q8, C12.18Q16, C8.2(C3×Q8), (C2×C16).3C6, (C2×C48).9C2, C2.2(C3×D16), (C2×C6).51D8, C4.1(C3×Q16), C2.2(C3×Q32), C2.D8.2C6, C24.75(C2×C4), C8.13(C2×C12), C12.56(C4⋊C4), (C2×C12).408D4, C6.13(C2.D8), C22.10(C3×D8), (C2×C24).396C22, C4.7(C3×C4⋊C4), C2.3(C3×C2.D8), (C2×C8).73(C2×C6), (C2×C4).62(C3×D4), (C3×C2.D8).9C2, SmallGroup(192,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C16⋊3C4
G = < a,b,c | a3=b16=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C16⋊3C4
►Regular action on 192 points
Generators in S192
(1 119 183)(2 120 184)(3 121 185)(4 122 186)(5 123 187)(6 124 188)(7 125 189)(8 126 190)(9 127 191)(10 128 192)(11 113 177)(12 114 178)(13 115 179)(14 116 180)(15 117 181)(16 118 182)(17 81 129)(18 82 130)(19 83 131)(20 84 132)(21 85 133)(22 86 134)(23 87 135)(24 88 136)(25 89 137)(26 90 138)(27 91 139)(28 92 140)(29 93 141)(30 94 142)(31 95 143)(32 96 144)(33 69 161)(34 70 162)(35 71 163)(36 72 164)(37 73 165)(38 74 166)(39 75 167)(40 76 168)(41 77 169)(42 78 170)(43 79 171)(44 80 172)(45 65 173)(46 66 174)(47 67 175)(48 68 176)(49 153 104)(50 154 105)(51 155 106)(52 156 107)(53 157 108)(54 158 109)(55 159 110)(56 160 111)(57 145 112)(58 146 97)(59 147 98)(60 148 99)(61 149 100)(62 150 101)(63 151 102)(64 152 103)
(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)
(1 131 103 168)(2 130 104 167)(3 129 105 166)(4 144 106 165)(5 143 107 164)(6 142 108 163)(7 141 109 162)(8 140 110 161)(9 139 111 176)(10 138 112 175)(11 137 97 174)(12 136 98 173)(13 135 99 172)(14 134 100 171)(15 133 101 170)(16 132 102 169)(17 50 38 121)(18 49 39 120)(19 64 40 119)(20 63 41 118)(21 62 42 117)(22 61 43 116)(23 60 44 115)(24 59 45 114)(25 58 46 113)(26 57 47 128)(27 56 48 127)(28 55 33 126)(29 54 34 125)(30 53 35 124)(31 52 36 123)(32 51 37 122)(65 178 88 147)(66 177 89 146)(67 192 90 145)(68 191 91 160)(69 190 92 159)(70 189 93 158)(71 188 94 157)(72 187 95 156)(73 186 96 155)(74 185 81 154)(75 184 82 153)(76 183 83 152)(77 182 84 151)(78 181 85 150)(79 180 86 149)(80 179 87 148)
G:=sub<Sym(192)| (1,119,183)(2,120,184)(3,121,185)(4,122,186)(5,123,187)(6,124,188)(7,125,189)(8,126,190)(9,127,191)(10,128,192)(11,113,177)(12,114,178)(13,115,179)(14,116,180)(15,117,181)(16,118,182)(17,81,129)(18,82,130)(19,83,131)(20,84,132)(21,85,133)(22,86,134)(23,87,135)(24,88,136)(25,89,137)(26,90,138)(27,91,139)(28,92,140)(29,93,141)(30,94,142)(31,95,143)(32,96,144)(33,69,161)(34,70,162)(35,71,163)(36,72,164)(37,73,165)(38,74,166)(39,75,167)(40,76,168)(41,77,169)(42,78,170)(43,79,171)(44,80,172)(45,65,173)(46,66,174)(47,67,175)(48,68,176)(49,153,104)(50,154,105)(51,155,106)(52,156,107)(53,157,108)(54,158,109)(55,159,110)(56,160,111)(57,145,112)(58,146,97)(59,147,98)(60,148,99)(61,149,100)(62,150,101)(63,151,102)(64,152,103), (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), (1,131,103,168)(2,130,104,167)(3,129,105,166)(4,144,106,165)(5,143,107,164)(6,142,108,163)(7,141,109,162)(8,140,110,161)(9,139,111,176)(10,138,112,175)(11,137,97,174)(12,136,98,173)(13,135,99,172)(14,134,100,171)(15,133,101,170)(16,132,102,169)(17,50,38,121)(18,49,39,120)(19,64,40,119)(20,63,41,118)(21,62,42,117)(22,61,43,116)(23,60,44,115)(24,59,45,114)(25,58,46,113)(26,57,47,128)(27,56,48,127)(28,55,33,126)(29,54,34,125)(30,53,35,124)(31,52,36,123)(32,51,37,122)(65,178,88,147)(66,177,89,146)(67,192,90,145)(68,191,91,160)(69,190,92,159)(70,189,93,158)(71,188,94,157)(72,187,95,156)(73,186,96,155)(74,185,81,154)(75,184,82,153)(76,183,83,152)(77,182,84,151)(78,181,85,150)(79,180,86,149)(80,179,87,148)>;
G:=Group( (1,119,183)(2,120,184)(3,121,185)(4,122,186)(5,123,187)(6,124,188)(7,125,189)(8,126,190)(9,127,191)(10,128,192)(11,113,177)(12,114,178)(13,115,179)(14,116,180)(15,117,181)(16,118,182)(17,81,129)(18,82,130)(19,83,131)(20,84,132)(21,85,133)(22,86,134)(23,87,135)(24,88,136)(25,89,137)(26,90,138)(27,91,139)(28,92,140)(29,93,141)(30,94,142)(31,95,143)(32,96,144)(33,69,161)(34,70,162)(35,71,163)(36,72,164)(37,73,165)(38,74,166)(39,75,167)(40,76,168)(41,77,169)(42,78,170)(43,79,171)(44,80,172)(45,65,173)(46,66,174)(47,67,175)(48,68,176)(49,153,104)(50,154,105)(51,155,106)(52,156,107)(53,157,108)(54,158,109)(55,159,110)(56,160,111)(57,145,112)(58,146,97)(59,147,98)(60,148,99)(61,149,100)(62,150,101)(63,151,102)(64,152,103), (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), (1,131,103,168)(2,130,104,167)(3,129,105,166)(4,144,106,165)(5,143,107,164)(6,142,108,163)(7,141,109,162)(8,140,110,161)(9,139,111,176)(10,138,112,175)(11,137,97,174)(12,136,98,173)(13,135,99,172)(14,134,100,171)(15,133,101,170)(16,132,102,169)(17,50,38,121)(18,49,39,120)(19,64,40,119)(20,63,41,118)(21,62,42,117)(22,61,43,116)(23,60,44,115)(24,59,45,114)(25,58,46,113)(26,57,47,128)(27,56,48,127)(28,55,33,126)(29,54,34,125)(30,53,35,124)(31,52,36,123)(32,51,37,122)(65,178,88,147)(66,177,89,146)(67,192,90,145)(68,191,91,160)(69,190,92,159)(70,189,93,158)(71,188,94,157)(72,187,95,156)(73,186,96,155)(74,185,81,154)(75,184,82,153)(76,183,83,152)(77,182,84,151)(78,181,85,150)(79,180,86,149)(80,179,87,148) );
G=PermutationGroup([[(1,119,183),(2,120,184),(3,121,185),(4,122,186),(5,123,187),(6,124,188),(7,125,189),(8,126,190),(9,127,191),(10,128,192),(11,113,177),(12,114,178),(13,115,179),(14,116,180),(15,117,181),(16,118,182),(17,81,129),(18,82,130),(19,83,131),(20,84,132),(21,85,133),(22,86,134),(23,87,135),(24,88,136),(25,89,137),(26,90,138),(27,91,139),(28,92,140),(29,93,141),(30,94,142),(31,95,143),(32,96,144),(33,69,161),(34,70,162),(35,71,163),(36,72,164),(37,73,165),(38,74,166),(39,75,167),(40,76,168),(41,77,169),(42,78,170),(43,79,171),(44,80,172),(45,65,173),(46,66,174),(47,67,175),(48,68,176),(49,153,104),(50,154,105),(51,155,106),(52,156,107),(53,157,108),(54,158,109),(55,159,110),(56,160,111),(57,145,112),(58,146,97),(59,147,98),(60,148,99),(61,149,100),(62,150,101),(63,151,102),(64,152,103)], [(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)], [(1,131,103,168),(2,130,104,167),(3,129,105,166),(4,144,106,165),(5,143,107,164),(6,142,108,163),(7,141,109,162),(8,140,110,161),(9,139,111,176),(10,138,112,175),(11,137,97,174),(12,136,98,173),(13,135,99,172),(14,134,100,171),(15,133,101,170),(16,132,102,169),(17,50,38,121),(18,49,39,120),(19,64,40,119),(20,63,41,118),(21,62,42,117),(22,61,43,116),(23,60,44,115),(24,59,45,114),(25,58,46,113),(26,57,47,128),(27,56,48,127),(28,55,33,126),(29,54,34,125),(30,53,35,124),(31,52,36,123),(32,51,37,122),(65,178,88,147),(66,177,89,146),(67,192,90,145),(68,191,91,160),(69,190,92,159),(70,189,93,158),(71,188,94,157),(72,187,95,156),(73,186,96,155),(74,185,81,154),(75,184,82,153),(76,183,83,152),(77,182,84,151),(78,181,85,150),(79,180,86,149),(80,179,87,148)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | Q16 | D8 | C3×Q8 | C3×D4 | D16 | Q32 | C3×Q16 | C3×D8 | C3×D16 | C3×Q32 |
| kernel | C3×C16⋊3C4 | C3×C2.D8 | C2×C48 | C16⋊3C4 | C48 | C2.D8 | C2×C16 | C16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C16⋊3C4 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 61 | 0 |
| 0 | 0 | 61 |
| 1 | 0 | 0 |
| 0 | 73 | 45 |
| 0 | 26 | 28 |
| 22 | 0 | 0 |
| 0 | 31 | 68 |
| 0 | 3 | 66 |
G:=sub<GL(3,GF(97))| [1,0,0,0,61,0,0,0,61],[1,0,0,0,73,26,0,45,28],[22,0,0,0,31,3,0,68,66] >;
C3×C16⋊3C4 in GAP, Magma, Sage, TeX
C_3\times C_{16}\rtimes_3C_4 % in TeX
G:=Group("C3xC16:3C4"); // GroupNames label
G:=SmallGroup(192,172);
// by ID
G=gap.SmallGroup(192,172);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,428,1683,360,6053,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^16=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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