direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4⋊Q16, C12⋊6Q16, C24.67D4, (C4×C8).7C6, C4⋊1(C3×Q16), C8.8(C3×D4), C4.3(C6×D4), C4⋊Q8.9C6, (C4×C24).18C2, (C2×Q16).3C6, C2.10(C6×Q16), C6.57(C2×Q16), (C2×C12).423D4, C12.310(C2×D4), C42.81(C2×C6), (C6×Q16).10C2, C6.44(C4⋊1D4), C22.115(C6×D4), (C2×C12).950C23, (C4×C12).365C22, (C2×C24).402C22, (C6×Q8).177C22, (C2×C8).80(C2×C6), (C2×C4).79(C3×D4), C2.7(C3×C4⋊1D4), (C3×C4⋊Q8).24C2, (C2×C6).671(C2×D4), (C2×Q8).22(C2×C6), (C2×C4).125(C22×C6), SmallGroup(192,927)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊Q16
G = < a,b,c,d | a3=b4=c8=1, d2=c4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 194 in 122 conjugacy classes, 66 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C24, C2×C12, C2×C12, C2×C12, C3×Q8, C4×C8, C4⋊Q8, C2×Q16, C4×C12, C3×C4⋊C4, C2×C24, C3×Q16, C6×Q8, C4⋊Q16, C4×C24, C3×C4⋊Q8, C6×Q16, C3×C4⋊Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C3×D4, C22×C6, C4⋊1D4, C2×Q16, C3×Q16, C6×D4, C4⋊Q16, C3×C4⋊1D4, C6×Q16, C3×C4⋊Q16
►Regular action on 192 points
Generators in S192
(1 67 19)(2 68 20)(3 69 21)(4 70 22)(5 71 23)(6 72 24)(7 65 17)(8 66 18)(9 25 61)(10 26 62)(11 27 63)(12 28 64)(13 29 57)(14 30 58)(15 31 59)(16 32 60)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 101 89)(42 102 90)(43 103 91)(44 104 92)(45 97 93)(46 98 94)(47 99 95)(48 100 96)(49 188 144)(50 189 137)(51 190 138)(52 191 139)(53 192 140)(54 185 141)(55 186 142)(56 187 143)(105 113 157)(106 114 158)(107 115 159)(108 116 160)(109 117 153)(110 118 154)(111 119 155)(112 120 156)(121 165 150)(122 166 151)(123 167 152)(124 168 145)(125 161 146)(126 162 147)(127 163 148)(128 164 149)(129 173 181)(130 174 182)(131 175 183)(132 176 184)(133 169 177)(134 170 178)(135 171 179)(136 172 180)
(1 45 79 25)(2 46 80 26)(3 47 73 27)(4 48 74 28)(5 41 75 29)(6 42 76 30)(7 43 77 31)(8 44 78 32)(9 19 93 39)(10 20 94 40)(11 21 95 33)(12 22 96 34)(13 23 89 35)(14 24 90 36)(15 17 91 37)(16 18 92 38)(49 116 147 178)(50 117 148 179)(51 118 149 180)(52 119 150 181)(53 120 151 182)(54 113 152 183)(55 114 145 184)(56 115 146 177)(57 71 101 83)(58 72 102 84)(59 65 103 85)(60 66 104 86)(61 67 97 87)(62 68 98 88)(63 69 99 81)(64 70 100 82)(105 167 175 141)(106 168 176 142)(107 161 169 143)(108 162 170 144)(109 163 171 137)(110 164 172 138)(111 165 173 139)(112 166 174 140)(121 129 191 155)(122 130 192 156)(123 131 185 157)(124 132 186 158)(125 133 187 159)(126 134 188 160)(127 135 189 153)(128 136 190 154)
(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)
(1 111 5 107)(2 110 6 106)(3 109 7 105)(4 108 8 112)(9 121 13 125)(10 128 14 124)(11 127 15 123)(12 126 16 122)(17 157 21 153)(18 156 22 160)(19 155 23 159)(20 154 24 158)(25 165 29 161)(26 164 30 168)(27 163 31 167)(28 162 32 166)(33 135 37 131)(34 134 38 130)(35 133 39 129)(36 132 40 136)(41 143 45 139)(42 142 46 138)(43 141 47 137)(44 140 48 144)(49 104 53 100)(50 103 54 99)(51 102 55 98)(52 101 56 97)(57 146 61 150)(58 145 62 149)(59 152 63 148)(60 151 64 147)(65 113 69 117)(66 120 70 116)(67 119 71 115)(68 118 72 114)(73 171 77 175)(74 170 78 174)(75 169 79 173)(76 176 80 172)(81 179 85 183)(82 178 86 182)(83 177 87 181)(84 184 88 180)(89 187 93 191)(90 186 94 190)(91 185 95 189)(92 192 96 188)
G:=sub<Sym(192)| (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,25,61)(10,26,62)(11,27,63)(12,28,64)(13,29,57)(14,30,58)(15,31,59)(16,32,60)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,101,89)(42,102,90)(43,103,91)(44,104,92)(45,97,93)(46,98,94)(47,99,95)(48,100,96)(49,188,144)(50,189,137)(51,190,138)(52,191,139)(53,192,140)(54,185,141)(55,186,142)(56,187,143)(105,113,157)(106,114,158)(107,115,159)(108,116,160)(109,117,153)(110,118,154)(111,119,155)(112,120,156)(121,165,150)(122,166,151)(123,167,152)(124,168,145)(125,161,146)(126,162,147)(127,163,148)(128,164,149)(129,173,181)(130,174,182)(131,175,183)(132,176,184)(133,169,177)(134,170,178)(135,171,179)(136,172,180), (1,45,79,25)(2,46,80,26)(3,47,73,27)(4,48,74,28)(5,41,75,29)(6,42,76,30)(7,43,77,31)(8,44,78,32)(9,19,93,39)(10,20,94,40)(11,21,95,33)(12,22,96,34)(13,23,89,35)(14,24,90,36)(15,17,91,37)(16,18,92,38)(49,116,147,178)(50,117,148,179)(51,118,149,180)(52,119,150,181)(53,120,151,182)(54,113,152,183)(55,114,145,184)(56,115,146,177)(57,71,101,83)(58,72,102,84)(59,65,103,85)(60,66,104,86)(61,67,97,87)(62,68,98,88)(63,69,99,81)(64,70,100,82)(105,167,175,141)(106,168,176,142)(107,161,169,143)(108,162,170,144)(109,163,171,137)(110,164,172,138)(111,165,173,139)(112,166,174,140)(121,129,191,155)(122,130,192,156)(123,131,185,157)(124,132,186,158)(125,133,187,159)(126,134,188,160)(127,135,189,153)(128,136,190,154), (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), (1,111,5,107)(2,110,6,106)(3,109,7,105)(4,108,8,112)(9,121,13,125)(10,128,14,124)(11,127,15,123)(12,126,16,122)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,165,29,161)(26,164,30,168)(27,163,31,167)(28,162,32,166)(33,135,37,131)(34,134,38,130)(35,133,39,129)(36,132,40,136)(41,143,45,139)(42,142,46,138)(43,141,47,137)(44,140,48,144)(49,104,53,100)(50,103,54,99)(51,102,55,98)(52,101,56,97)(57,146,61,150)(58,145,62,149)(59,152,63,148)(60,151,64,147)(65,113,69,117)(66,120,70,116)(67,119,71,115)(68,118,72,114)(73,171,77,175)(74,170,78,174)(75,169,79,173)(76,176,80,172)(81,179,85,183)(82,178,86,182)(83,177,87,181)(84,184,88,180)(89,187,93,191)(90,186,94,190)(91,185,95,189)(92,192,96,188)>;
G:=Group( (1,67,19)(2,68,20)(3,69,21)(4,70,22)(5,71,23)(6,72,24)(7,65,17)(8,66,18)(9,25,61)(10,26,62)(11,27,63)(12,28,64)(13,29,57)(14,30,58)(15,31,59)(16,32,60)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,101,89)(42,102,90)(43,103,91)(44,104,92)(45,97,93)(46,98,94)(47,99,95)(48,100,96)(49,188,144)(50,189,137)(51,190,138)(52,191,139)(53,192,140)(54,185,141)(55,186,142)(56,187,143)(105,113,157)(106,114,158)(107,115,159)(108,116,160)(109,117,153)(110,118,154)(111,119,155)(112,120,156)(121,165,150)(122,166,151)(123,167,152)(124,168,145)(125,161,146)(126,162,147)(127,163,148)(128,164,149)(129,173,181)(130,174,182)(131,175,183)(132,176,184)(133,169,177)(134,170,178)(135,171,179)(136,172,180), (1,45,79,25)(2,46,80,26)(3,47,73,27)(4,48,74,28)(5,41,75,29)(6,42,76,30)(7,43,77,31)(8,44,78,32)(9,19,93,39)(10,20,94,40)(11,21,95,33)(12,22,96,34)(13,23,89,35)(14,24,90,36)(15,17,91,37)(16,18,92,38)(49,116,147,178)(50,117,148,179)(51,118,149,180)(52,119,150,181)(53,120,151,182)(54,113,152,183)(55,114,145,184)(56,115,146,177)(57,71,101,83)(58,72,102,84)(59,65,103,85)(60,66,104,86)(61,67,97,87)(62,68,98,88)(63,69,99,81)(64,70,100,82)(105,167,175,141)(106,168,176,142)(107,161,169,143)(108,162,170,144)(109,163,171,137)(110,164,172,138)(111,165,173,139)(112,166,174,140)(121,129,191,155)(122,130,192,156)(123,131,185,157)(124,132,186,158)(125,133,187,159)(126,134,188,160)(127,135,189,153)(128,136,190,154), (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), (1,111,5,107)(2,110,6,106)(3,109,7,105)(4,108,8,112)(9,121,13,125)(10,128,14,124)(11,127,15,123)(12,126,16,122)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,165,29,161)(26,164,30,168)(27,163,31,167)(28,162,32,166)(33,135,37,131)(34,134,38,130)(35,133,39,129)(36,132,40,136)(41,143,45,139)(42,142,46,138)(43,141,47,137)(44,140,48,144)(49,104,53,100)(50,103,54,99)(51,102,55,98)(52,101,56,97)(57,146,61,150)(58,145,62,149)(59,152,63,148)(60,151,64,147)(65,113,69,117)(66,120,70,116)(67,119,71,115)(68,118,72,114)(73,171,77,175)(74,170,78,174)(75,169,79,173)(76,176,80,172)(81,179,85,183)(82,178,86,182)(83,177,87,181)(84,184,88,180)(89,187,93,191)(90,186,94,190)(91,185,95,189)(92,192,96,188) );
G=PermutationGroup([[(1,67,19),(2,68,20),(3,69,21),(4,70,22),(5,71,23),(6,72,24),(7,65,17),(8,66,18),(9,25,61),(10,26,62),(11,27,63),(12,28,64),(13,29,57),(14,30,58),(15,31,59),(16,32,60),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,101,89),(42,102,90),(43,103,91),(44,104,92),(45,97,93),(46,98,94),(47,99,95),(48,100,96),(49,188,144),(50,189,137),(51,190,138),(52,191,139),(53,192,140),(54,185,141),(55,186,142),(56,187,143),(105,113,157),(106,114,158),(107,115,159),(108,116,160),(109,117,153),(110,118,154),(111,119,155),(112,120,156),(121,165,150),(122,166,151),(123,167,152),(124,168,145),(125,161,146),(126,162,147),(127,163,148),(128,164,149),(129,173,181),(130,174,182),(131,175,183),(132,176,184),(133,169,177),(134,170,178),(135,171,179),(136,172,180)], [(1,45,79,25),(2,46,80,26),(3,47,73,27),(4,48,74,28),(5,41,75,29),(6,42,76,30),(7,43,77,31),(8,44,78,32),(9,19,93,39),(10,20,94,40),(11,21,95,33),(12,22,96,34),(13,23,89,35),(14,24,90,36),(15,17,91,37),(16,18,92,38),(49,116,147,178),(50,117,148,179),(51,118,149,180),(52,119,150,181),(53,120,151,182),(54,113,152,183),(55,114,145,184),(56,115,146,177),(57,71,101,83),(58,72,102,84),(59,65,103,85),(60,66,104,86),(61,67,97,87),(62,68,98,88),(63,69,99,81),(64,70,100,82),(105,167,175,141),(106,168,176,142),(107,161,169,143),(108,162,170,144),(109,163,171,137),(110,164,172,138),(111,165,173,139),(112,166,174,140),(121,129,191,155),(122,130,192,156),(123,131,185,157),(124,132,186,158),(125,133,187,159),(126,134,188,160),(127,135,189,153),(128,136,190,154)], [(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)], [(1,111,5,107),(2,110,6,106),(3,109,7,105),(4,108,8,112),(9,121,13,125),(10,128,14,124),(11,127,15,123),(12,126,16,122),(17,157,21,153),(18,156,22,160),(19,155,23,159),(20,154,24,158),(25,165,29,161),(26,164,30,168),(27,163,31,167),(28,162,32,166),(33,135,37,131),(34,134,38,130),(35,133,39,129),(36,132,40,136),(41,143,45,139),(42,142,46,138),(43,141,47,137),(44,140,48,144),(49,104,53,100),(50,103,54,99),(51,102,55,98),(52,101,56,97),(57,146,61,150),(58,145,62,149),(59,152,63,148),(60,151,64,147),(65,113,69,117),(66,120,70,116),(67,119,71,115),(68,118,72,114),(73,171,77,175),(74,170,78,174),(75,169,79,173),(76,176,80,172),(81,179,85,183),(82,178,86,182),(83,177,87,181),(84,184,88,180),(89,187,93,191),(90,186,94,190),(91,185,95,189),(92,192,96,188)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12L | 12M | ··· | 12T | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | Q16 | C3×D4 | C3×D4 | C3×Q16 |
| kernel | C3×C4⋊Q16 | C4×C24 | C3×C4⋊Q8 | C6×Q16 | C4⋊Q16 | C4×C8 | C4⋊Q8 | C2×Q16 | C24 | C2×C12 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 2 | 8 | 8 | 4 | 16 |
Matrix representation of C3×C4⋊Q16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 57 | 57 |
| 23 | 28 | 0 | 0 |
| 28 | 50 | 0 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 27 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[0,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[16,16,0,0,57,16,0,0,0,0,57,57,0,0,16,57],[23,28,0,0,28,50,0,0,0,0,0,27,0,0,27,0] >;
C3×C4⋊Q16 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes Q_{16} % in TeX
G:=Group("C3xC4:Q16"); // GroupNames label
G:=SmallGroup(192,927);
// by ID
G=gap.SmallGroup(192,927);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094,268,4204,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations