direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C3⋊C16, C12⋊2C16, C24.3C8, C12.13C42, C42.13Dic3, C3⋊1(C4×C16), C8.5(C3⋊C8), C6.3(C4×C8), C8.38(C4×S3), C6.6(C2×C16), (C4×C8).18S3, (C4×C12).17C4, (C2×C24).23C4, C24.59(C2×C4), C12.40(C2×C8), (C2×C12).10C8, (C4×C24).24C2, (C2×C8).329D6, C4.13(C4×Dic3), (C2×C8).17Dic3, (C2×C24).413C22, C2.2(C4×C3⋊C8), C2.1(C2×C3⋊C16), C4.12(C2×C3⋊C8), (C2×C4).7(C3⋊C8), C22.7(C2×C3⋊C8), (C2×C3⋊C16).11C2, (C2×C6).25(C2×C8), (C2×C12).309(C2×C4), (C2×C4).90(C2×Dic3), SmallGroup(192,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C4×C3⋊C16 |
Generators and relations for C4×C3⋊C16
G = < a,b,c | a4=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×C3⋊C16
►Regular action on 192 points
Generators in S192
(1 111 33 192)(2 112 34 177)(3 97 35 178)(4 98 36 179)(5 99 37 180)(6 100 38 181)(7 101 39 182)(8 102 40 183)(9 103 41 184)(10 104 42 185)(11 105 43 186)(12 106 44 187)(13 107 45 188)(14 108 46 189)(15 109 47 190)(16 110 48 191)(17 117 50 80)(18 118 51 65)(19 119 52 66)(20 120 53 67)(21 121 54 68)(22 122 55 69)(23 123 56 70)(24 124 57 71)(25 125 58 72)(26 126 59 73)(27 127 60 74)(28 128 61 75)(29 113 62 76)(30 114 63 77)(31 115 64 78)(32 116 49 79)(81 146 129 166)(82 147 130 167)(83 148 131 168)(84 149 132 169)(85 150 133 170)(86 151 134 171)(87 152 135 172)(88 153 136 173)(89 154 137 174)(90 155 138 175)(91 156 139 176)(92 157 140 161)(93 158 141 162)(94 159 142 163)(95 160 143 164)(96 145 144 165)
(1 121 85)(2 86 122)(3 123 87)(4 88 124)(5 125 89)(6 90 126)(7 127 91)(8 92 128)(9 113 93)(10 94 114)(11 115 95)(12 96 116)(13 117 81)(14 82 118)(15 119 83)(16 84 120)(17 166 188)(18 189 167)(19 168 190)(20 191 169)(21 170 192)(22 177 171)(23 172 178)(24 179 173)(25 174 180)(26 181 175)(27 176 182)(28 183 161)(29 162 184)(30 185 163)(31 164 186)(32 187 165)(33 68 133)(34 134 69)(35 70 135)(36 136 71)(37 72 137)(38 138 73)(39 74 139)(40 140 75)(41 76 141)(42 142 77)(43 78 143)(44 144 79)(45 80 129)(46 130 65)(47 66 131)(48 132 67)(49 106 145)(50 146 107)(51 108 147)(52 148 109)(53 110 149)(54 150 111)(55 112 151)(56 152 97)(57 98 153)(58 154 99)(59 100 155)(60 156 101)(61 102 157)(62 158 103)(63 104 159)(64 160 105)
(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)| (1,111,33,192)(2,112,34,177)(3,97,35,178)(4,98,36,179)(5,99,37,180)(6,100,38,181)(7,101,39,182)(8,102,40,183)(9,103,41,184)(10,104,42,185)(11,105,43,186)(12,106,44,187)(13,107,45,188)(14,108,46,189)(15,109,47,190)(16,110,48,191)(17,117,50,80)(18,118,51,65)(19,119,52,66)(20,120,53,67)(21,121,54,68)(22,122,55,69)(23,123,56,70)(24,124,57,71)(25,125,58,72)(26,126,59,73)(27,127,60,74)(28,128,61,75)(29,113,62,76)(30,114,63,77)(31,115,64,78)(32,116,49,79)(81,146,129,166)(82,147,130,167)(83,148,131,168)(84,149,132,169)(85,150,133,170)(86,151,134,171)(87,152,135,172)(88,153,136,173)(89,154,137,174)(90,155,138,175)(91,156,139,176)(92,157,140,161)(93,158,141,162)(94,159,142,163)(95,160,143,164)(96,145,144,165), (1,121,85)(2,86,122)(3,123,87)(4,88,124)(5,125,89)(6,90,126)(7,127,91)(8,92,128)(9,113,93)(10,94,114)(11,115,95)(12,96,116)(13,117,81)(14,82,118)(15,119,83)(16,84,120)(17,166,188)(18,189,167)(19,168,190)(20,191,169)(21,170,192)(22,177,171)(23,172,178)(24,179,173)(25,174,180)(26,181,175)(27,176,182)(28,183,161)(29,162,184)(30,185,163)(31,164,186)(32,187,165)(33,68,133)(34,134,69)(35,70,135)(36,136,71)(37,72,137)(38,138,73)(39,74,139)(40,140,75)(41,76,141)(42,142,77)(43,78,143)(44,144,79)(45,80,129)(46,130,65)(47,66,131)(48,132,67)(49,106,145)(50,146,107)(51,108,147)(52,148,109)(53,110,149)(54,150,111)(55,112,151)(56,152,97)(57,98,153)(58,154,99)(59,100,155)(60,156,101)(61,102,157)(62,158,103)(63,104,159)(64,160,105), (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( (1,111,33,192)(2,112,34,177)(3,97,35,178)(4,98,36,179)(5,99,37,180)(6,100,38,181)(7,101,39,182)(8,102,40,183)(9,103,41,184)(10,104,42,185)(11,105,43,186)(12,106,44,187)(13,107,45,188)(14,108,46,189)(15,109,47,190)(16,110,48,191)(17,117,50,80)(18,118,51,65)(19,119,52,66)(20,120,53,67)(21,121,54,68)(22,122,55,69)(23,123,56,70)(24,124,57,71)(25,125,58,72)(26,126,59,73)(27,127,60,74)(28,128,61,75)(29,113,62,76)(30,114,63,77)(31,115,64,78)(32,116,49,79)(81,146,129,166)(82,147,130,167)(83,148,131,168)(84,149,132,169)(85,150,133,170)(86,151,134,171)(87,152,135,172)(88,153,136,173)(89,154,137,174)(90,155,138,175)(91,156,139,176)(92,157,140,161)(93,158,141,162)(94,159,142,163)(95,160,143,164)(96,145,144,165), (1,121,85)(2,86,122)(3,123,87)(4,88,124)(5,125,89)(6,90,126)(7,127,91)(8,92,128)(9,113,93)(10,94,114)(11,115,95)(12,96,116)(13,117,81)(14,82,118)(15,119,83)(16,84,120)(17,166,188)(18,189,167)(19,168,190)(20,191,169)(21,170,192)(22,177,171)(23,172,178)(24,179,173)(25,174,180)(26,181,175)(27,176,182)(28,183,161)(29,162,184)(30,185,163)(31,164,186)(32,187,165)(33,68,133)(34,134,69)(35,70,135)(36,136,71)(37,72,137)(38,138,73)(39,74,139)(40,140,75)(41,76,141)(42,142,77)(43,78,143)(44,144,79)(45,80,129)(46,130,65)(47,66,131)(48,132,67)(49,106,145)(50,146,107)(51,108,147)(52,148,109)(53,110,149)(54,150,111)(55,112,151)(56,152,97)(57,98,153)(58,154,99)(59,100,155)(60,156,101)(61,102,157)(62,158,103)(63,104,159)(64,160,105), (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([[(1,111,33,192),(2,112,34,177),(3,97,35,178),(4,98,36,179),(5,99,37,180),(6,100,38,181),(7,101,39,182),(8,102,40,183),(9,103,41,184),(10,104,42,185),(11,105,43,186),(12,106,44,187),(13,107,45,188),(14,108,46,189),(15,109,47,190),(16,110,48,191),(17,117,50,80),(18,118,51,65),(19,119,52,66),(20,120,53,67),(21,121,54,68),(22,122,55,69),(23,123,56,70),(24,124,57,71),(25,125,58,72),(26,126,59,73),(27,127,60,74),(28,128,61,75),(29,113,62,76),(30,114,63,77),(31,115,64,78),(32,116,49,79),(81,146,129,166),(82,147,130,167),(83,148,131,168),(84,149,132,169),(85,150,133,170),(86,151,134,171),(87,152,135,172),(88,153,136,173),(89,154,137,174),(90,155,138,175),(91,156,139,176),(92,157,140,161),(93,158,141,162),(94,159,142,163),(95,160,143,164),(96,145,144,165)], [(1,121,85),(2,86,122),(3,123,87),(4,88,124),(5,125,89),(6,90,126),(7,127,91),(8,92,128),(9,113,93),(10,94,114),(11,115,95),(12,96,116),(13,117,81),(14,82,118),(15,119,83),(16,84,120),(17,166,188),(18,189,167),(19,168,190),(20,191,169),(21,170,192),(22,177,171),(23,172,178),(24,179,173),(25,174,180),(26,181,175),(27,176,182),(28,183,161),(29,162,184),(30,185,163),(31,164,186),(32,187,165),(33,68,133),(34,134,69),(35,70,135),(36,136,71),(37,72,137),(38,138,73),(39,74,139),(40,140,75),(41,76,141),(42,142,77),(43,78,143),(44,144,79),(45,80,129),(46,130,65),(47,66,131),(48,132,67),(49,106,145),(50,146,107),(51,108,147),(52,148,109),(53,110,149),(54,150,111),(55,112,151),(56,152,97),(57,98,153),(58,154,99),(59,100,155),(60,156,101),(61,102,157),(62,158,103),(63,104,159),(64,160,105)], [(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 | 2B | 2C | 3 | 4A | ··· | 4L | 6A | 6B | 6C | 8A | ··· | 8P | 12A | ··· | 12L | 16A | ··· | 16AF | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | C16 | S3 | Dic3 | Dic3 | D6 | C3⋊C8 | C4×S3 | C3⋊C8 | C3⋊C16 |
| kernel | C4×C3⋊C16 | C2×C3⋊C16 | C4×C24 | C3⋊C16 | C4×C12 | C2×C24 | C24 | C2×C12 | C12 | C4×C8 | C42 | C2×C8 | C2×C8 | C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 32 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C4×C3⋊C16 ►in GL3(𝔽97) generated by
| 75 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 1 | 0 | 0 |
| 0 | 0 | 96 |
| 0 | 1 | 96 |
| 1 | 0 | 0 |
| 0 | 42 | 62 |
| 0 | 7 | 55 |
G:=sub<GL(3,GF(97))| [75,0,0,0,96,0,0,0,96],[1,0,0,0,0,1,0,96,96],[1,0,0,0,42,7,0,62,55] >;
C4×C3⋊C16 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes C_{16} % in TeX
G:=Group("C4xC3:C16"); // GroupNames label
G:=SmallGroup(192,19);
// by ID
G=gap.SmallGroup(192,19);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,64,100,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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