direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4⋊C16, C4⋊C48, C12⋊3C16, C24.23Q8, C24.108D4, C42.7C12, C6.9M5(2), C12.38M4(2), (C4×C8).2C6, C8.7(C3×Q8), (C2×C8).7C12, (C4×C24).4C2, (C2×C48).4C2, C2.2(C2×C48), (C2×C4).4C24, (C2×C16).2C6, C8.28(C3×D4), C6.14(C4⋊C8), (C4×C12).22C4, (C2×C12).13C8, C6.12(C2×C16), (C2×C24).17C4, C12.67(C4⋊C4), C2.3(C3×M5(2)), C22.10(C2×C24), C4.11(C3×M4(2)), (C2×C24).452C22, C2.2(C3×C4⋊C8), C4.18(C3×C4⋊C4), (C2×C6).41(C2×C8), (C2×C4).84(C2×C12), (C2×C8).106(C2×C6), (C2×C12).346(C2×C4), SmallGroup(192,169)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊C16
G = < a,b,c | a3=b4=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C4⋊C16
►Regular action on 192 points
Generators in S192
(1 148 29)(2 149 30)(3 150 31)(4 151 32)(5 152 17)(6 153 18)(7 154 19)(8 155 20)(9 156 21)(10 157 22)(11 158 23)(12 159 24)(13 160 25)(14 145 26)(15 146 27)(16 147 28)(33 78 180)(34 79 181)(35 80 182)(36 65 183)(37 66 184)(38 67 185)(39 68 186)(40 69 187)(41 70 188)(42 71 189)(43 72 190)(44 73 191)(45 74 192)(46 75 177)(47 76 178)(48 77 179)(49 167 144)(50 168 129)(51 169 130)(52 170 131)(53 171 132)(54 172 133)(55 173 134)(56 174 135)(57 175 136)(58 176 137)(59 161 138)(60 162 139)(61 163 140)(62 164 141)(63 165 142)(64 166 143)(81 97 126)(82 98 127)(83 99 128)(84 100 113)(85 101 114)(86 102 115)(87 103 116)(88 104 117)(89 105 118)(90 106 119)(91 107 120)(92 108 121)(93 109 122)(94 110 123)(95 111 124)(96 112 125)
(1 123 70 143)(2 144 71 124)(3 125 72 129)(4 130 73 126)(5 127 74 131)(6 132 75 128)(7 113 76 133)(8 134 77 114)(9 115 78 135)(10 136 79 116)(11 117 80 137)(12 138 65 118)(13 119 66 139)(14 140 67 120)(15 121 68 141)(16 142 69 122)(17 98 45 170)(18 171 46 99)(19 100 47 172)(20 173 48 101)(21 102 33 174)(22 175 34 103)(23 104 35 176)(24 161 36 105)(25 106 37 162)(26 163 38 107)(27 108 39 164)(28 165 40 109)(29 110 41 166)(30 167 42 111)(31 112 43 168)(32 169 44 97)(49 189 95 149)(50 150 96 190)(51 191 81 151)(52 152 82 192)(53 177 83 153)(54 154 84 178)(55 179 85 155)(56 156 86 180)(57 181 87 157)(58 158 88 182)(59 183 89 159)(60 160 90 184)(61 185 91 145)(62 146 92 186)(63 187 93 147)(64 148 94 188)
(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)| (1,148,29)(2,149,30)(3,150,31)(4,151,32)(5,152,17)(6,153,18)(7,154,19)(8,155,20)(9,156,21)(10,157,22)(11,158,23)(12,159,24)(13,160,25)(14,145,26)(15,146,27)(16,147,28)(33,78,180)(34,79,181)(35,80,182)(36,65,183)(37,66,184)(38,67,185)(39,68,186)(40,69,187)(41,70,188)(42,71,189)(43,72,190)(44,73,191)(45,74,192)(46,75,177)(47,76,178)(48,77,179)(49,167,144)(50,168,129)(51,169,130)(52,170,131)(53,171,132)(54,172,133)(55,173,134)(56,174,135)(57,175,136)(58,176,137)(59,161,138)(60,162,139)(61,163,140)(62,164,141)(63,165,142)(64,166,143)(81,97,126)(82,98,127)(83,99,128)(84,100,113)(85,101,114)(86,102,115)(87,103,116)(88,104,117)(89,105,118)(90,106,119)(91,107,120)(92,108,121)(93,109,122)(94,110,123)(95,111,124)(96,112,125), (1,123,70,143)(2,144,71,124)(3,125,72,129)(4,130,73,126)(5,127,74,131)(6,132,75,128)(7,113,76,133)(8,134,77,114)(9,115,78,135)(10,136,79,116)(11,117,80,137)(12,138,65,118)(13,119,66,139)(14,140,67,120)(15,121,68,141)(16,142,69,122)(17,98,45,170)(18,171,46,99)(19,100,47,172)(20,173,48,101)(21,102,33,174)(22,175,34,103)(23,104,35,176)(24,161,36,105)(25,106,37,162)(26,163,38,107)(27,108,39,164)(28,165,40,109)(29,110,41,166)(30,167,42,111)(31,112,43,168)(32,169,44,97)(49,189,95,149)(50,150,96,190)(51,191,81,151)(52,152,82,192)(53,177,83,153)(54,154,84,178)(55,179,85,155)(56,156,86,180)(57,181,87,157)(58,158,88,182)(59,183,89,159)(60,160,90,184)(61,185,91,145)(62,146,92,186)(63,187,93,147)(64,148,94,188), (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( (1,148,29)(2,149,30)(3,150,31)(4,151,32)(5,152,17)(6,153,18)(7,154,19)(8,155,20)(9,156,21)(10,157,22)(11,158,23)(12,159,24)(13,160,25)(14,145,26)(15,146,27)(16,147,28)(33,78,180)(34,79,181)(35,80,182)(36,65,183)(37,66,184)(38,67,185)(39,68,186)(40,69,187)(41,70,188)(42,71,189)(43,72,190)(44,73,191)(45,74,192)(46,75,177)(47,76,178)(48,77,179)(49,167,144)(50,168,129)(51,169,130)(52,170,131)(53,171,132)(54,172,133)(55,173,134)(56,174,135)(57,175,136)(58,176,137)(59,161,138)(60,162,139)(61,163,140)(62,164,141)(63,165,142)(64,166,143)(81,97,126)(82,98,127)(83,99,128)(84,100,113)(85,101,114)(86,102,115)(87,103,116)(88,104,117)(89,105,118)(90,106,119)(91,107,120)(92,108,121)(93,109,122)(94,110,123)(95,111,124)(96,112,125), (1,123,70,143)(2,144,71,124)(3,125,72,129)(4,130,73,126)(5,127,74,131)(6,132,75,128)(7,113,76,133)(8,134,77,114)(9,115,78,135)(10,136,79,116)(11,117,80,137)(12,138,65,118)(13,119,66,139)(14,140,67,120)(15,121,68,141)(16,142,69,122)(17,98,45,170)(18,171,46,99)(19,100,47,172)(20,173,48,101)(21,102,33,174)(22,175,34,103)(23,104,35,176)(24,161,36,105)(25,106,37,162)(26,163,38,107)(27,108,39,164)(28,165,40,109)(29,110,41,166)(30,167,42,111)(31,112,43,168)(32,169,44,97)(49,189,95,149)(50,150,96,190)(51,191,81,151)(52,152,82,192)(53,177,83,153)(54,154,84,178)(55,179,85,155)(56,156,86,180)(57,181,87,157)(58,158,88,182)(59,183,89,159)(60,160,90,184)(61,185,91,145)(62,146,92,186)(63,187,93,147)(64,148,94,188), (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([[(1,148,29),(2,149,30),(3,150,31),(4,151,32),(5,152,17),(6,153,18),(7,154,19),(8,155,20),(9,156,21),(10,157,22),(11,158,23),(12,159,24),(13,160,25),(14,145,26),(15,146,27),(16,147,28),(33,78,180),(34,79,181),(35,80,182),(36,65,183),(37,66,184),(38,67,185),(39,68,186),(40,69,187),(41,70,188),(42,71,189),(43,72,190),(44,73,191),(45,74,192),(46,75,177),(47,76,178),(48,77,179),(49,167,144),(50,168,129),(51,169,130),(52,170,131),(53,171,132),(54,172,133),(55,173,134),(56,174,135),(57,175,136),(58,176,137),(59,161,138),(60,162,139),(61,163,140),(62,164,141),(63,165,142),(64,166,143),(81,97,126),(82,98,127),(83,99,128),(84,100,113),(85,101,114),(86,102,115),(87,103,116),(88,104,117),(89,105,118),(90,106,119),(91,107,120),(92,108,121),(93,109,122),(94,110,123),(95,111,124),(96,112,125)], [(1,123,70,143),(2,144,71,124),(3,125,72,129),(4,130,73,126),(5,127,74,131),(6,132,75,128),(7,113,76,133),(8,134,77,114),(9,115,78,135),(10,136,79,116),(11,117,80,137),(12,138,65,118),(13,119,66,139),(14,140,67,120),(15,121,68,141),(16,142,69,122),(17,98,45,170),(18,171,46,99),(19,100,47,172),(20,173,48,101),(21,102,33,174),(22,175,34,103),(23,104,35,176),(24,161,36,105),(25,106,37,162),(26,163,38,107),(27,108,39,164),(28,165,40,109),(29,110,41,166),(30,167,42,111),(31,112,43,168),(32,169,44,97),(49,189,95,149),(50,150,96,190),(51,191,81,151),(52,152,82,192),(53,177,83,153),(54,154,84,178),(55,179,85,155),(56,156,86,180),(57,181,87,157),(58,158,88,182),(59,183,89,159),(60,160,90,184),(61,185,91,145),(62,146,92,186),(63,187,93,147),(64,148,94,188)], [(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)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | ··· | 12P | 16A | ··· | 16P | 24A | ··· | 24P | 24Q | ··· | 24X | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C16 | C24 | C48 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | M5(2) | C3×M4(2) | C3×M5(2) |
| kernel | C3×C4⋊C16 | C4×C24 | C2×C48 | C4⋊C16 | C4×C12 | C2×C24 | C4×C8 | C2×C16 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C4 | C24 | C24 | C12 | C8 | C8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 4 | 4 | 16 | 16 | 32 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C4⋊C16 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 0 | 35 |
| 96 | 0 | 0 |
| 0 | 0 | 96 |
| 0 | 1 | 0 |
| 85 | 0 | 0 |
| 0 | 60 | 64 |
| 0 | 64 | 37 |
G:=sub<GL(3,GF(97))| [1,0,0,0,35,0,0,0,35],[96,0,0,0,0,1,0,96,0],[85,0,0,0,60,64,0,64,37] >;
C3×C4⋊C16 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes C_{16} % in TeX
G:=Group("C3xC4:C16"); // GroupNames label
G:=SmallGroup(192,169);
// by ID
G=gap.SmallGroup(192,169);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,92,102,124]);
// Polycyclic
G:=Group<a,b,c|a^3=b^4=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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