direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×Dic14, C28⋊3Q8, C42.3D7, C7⋊1(C4×Q8), C4.9(C4×D7), (C4×C28).5C2, C14.1(C2×Q8), C28.19(C2×C4), (C2×C4).72D14, C2.1(C4○D28), C14.1(C4○D4), Dic7⋊C4.7C2, C4⋊Dic7.13C2, C14.1(C22×C4), (C2×C14).9C23, (C4×Dic7).7C2, Dic7.1(C2×C4), C2.1(C2×Dic14), (C2×C28).84C22, C22.8(C22×D7), (C2×Dic14).10C2, (C2×Dic7).23C22, C2.4(C2×C4×D7), SmallGroup(224,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×Dic14
G = < a,b,c | a4=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 214 in 70 conjugacy classes, 45 normal (21 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, Q8, C14, C42, C42, C4⋊C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4×Q8, Dic14, C2×Dic7, C2×C28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4×C28, C2×Dic14, C4×Dic14
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D7, C22×C4, C2×Q8, C4○D4, D14, C4×Q8, Dic14, C4×D7, C22×D7, C2×Dic14, C2×C4×D7, C4○D28, C4×Dic14
►Regular action on 224 points
Generators in S224
(1 40 98 154)(2 41 99 155)(3 42 100 156)(4 43 101 157)(5 44 102 158)(6 45 103 159)(7 46 104 160)(8 47 105 161)(9 48 106 162)(10 49 107 163)(11 50 108 164)(12 51 109 165)(13 52 110 166)(14 53 111 167)(15 54 112 168)(16 55 85 141)(17 56 86 142)(18 29 87 143)(19 30 88 144)(20 31 89 145)(21 32 90 146)(22 33 91 147)(23 34 92 148)(24 35 93 149)(25 36 94 150)(26 37 95 151)(27 38 96 152)(28 39 97 153)(57 213 137 170)(58 214 138 171)(59 215 139 172)(60 216 140 173)(61 217 113 174)(62 218 114 175)(63 219 115 176)(64 220 116 177)(65 221 117 178)(66 222 118 179)(67 223 119 180)(68 224 120 181)(69 197 121 182)(70 198 122 183)(71 199 123 184)(72 200 124 185)(73 201 125 186)(74 202 126 187)(75 203 127 188)(76 204 128 189)(77 205 129 190)(78 206 130 191)(79 207 131 192)(80 208 132 193)(81 209 133 194)(82 210 134 195)(83 211 135 196)(84 212 136 169)
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)(197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 215 15 201)(2 214 16 200)(3 213 17 199)(4 212 18 198)(5 211 19 197)(6 210 20 224)(7 209 21 223)(8 208 22 222)(9 207 23 221)(10 206 24 220)(11 205 25 219)(12 204 26 218)(13 203 27 217)(14 202 28 216)(29 122 43 136)(30 121 44 135)(31 120 45 134)(32 119 46 133)(33 118 47 132)(34 117 48 131)(35 116 49 130)(36 115 50 129)(37 114 51 128)(38 113 52 127)(39 140 53 126)(40 139 54 125)(41 138 55 124)(42 137 56 123)(57 142 71 156)(58 141 72 155)(59 168 73 154)(60 167 74 153)(61 166 75 152)(62 165 76 151)(63 164 77 150)(64 163 78 149)(65 162 79 148)(66 161 80 147)(67 160 81 146)(68 159 82 145)(69 158 83 144)(70 157 84 143)(85 185 99 171)(86 184 100 170)(87 183 101 169)(88 182 102 196)(89 181 103 195)(90 180 104 194)(91 179 105 193)(92 178 106 192)(93 177 107 191)(94 176 108 190)(95 175 109 189)(96 174 110 188)(97 173 111 187)(98 172 112 186)
G:=sub<Sym(224)| (1,40,98,154)(2,41,99,155)(3,42,100,156)(4,43,101,157)(5,44,102,158)(6,45,103,159)(7,46,104,160)(8,47,105,161)(9,48,106,162)(10,49,107,163)(11,50,108,164)(12,51,109,165)(13,52,110,166)(14,53,111,167)(15,54,112,168)(16,55,85,141)(17,56,86,142)(18,29,87,143)(19,30,88,144)(20,31,89,145)(21,32,90,146)(22,33,91,147)(23,34,92,148)(24,35,93,149)(25,36,94,150)(26,37,95,151)(27,38,96,152)(28,39,97,153)(57,213,137,170)(58,214,138,171)(59,215,139,172)(60,216,140,173)(61,217,113,174)(62,218,114,175)(63,219,115,176)(64,220,116,177)(65,221,117,178)(66,222,118,179)(67,223,119,180)(68,224,120,181)(69,197,121,182)(70,198,122,183)(71,199,123,184)(72,200,124,185)(73,201,125,186)(74,202,126,187)(75,203,127,188)(76,204,128,189)(77,205,129,190)(78,206,130,191)(79,207,131,192)(80,208,132,193)(81,209,133,194)(82,210,134,195)(83,211,135,196)(84,212,136,169), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,215,15,201)(2,214,16,200)(3,213,17,199)(4,212,18,198)(5,211,19,197)(6,210,20,224)(7,209,21,223)(8,208,22,222)(9,207,23,221)(10,206,24,220)(11,205,25,219)(12,204,26,218)(13,203,27,217)(14,202,28,216)(29,122,43,136)(30,121,44,135)(31,120,45,134)(32,119,46,133)(33,118,47,132)(34,117,48,131)(35,116,49,130)(36,115,50,129)(37,114,51,128)(38,113,52,127)(39,140,53,126)(40,139,54,125)(41,138,55,124)(42,137,56,123)(57,142,71,156)(58,141,72,155)(59,168,73,154)(60,167,74,153)(61,166,75,152)(62,165,76,151)(63,164,77,150)(64,163,78,149)(65,162,79,148)(66,161,80,147)(67,160,81,146)(68,159,82,145)(69,158,83,144)(70,157,84,143)(85,185,99,171)(86,184,100,170)(87,183,101,169)(88,182,102,196)(89,181,103,195)(90,180,104,194)(91,179,105,193)(92,178,106,192)(93,177,107,191)(94,176,108,190)(95,175,109,189)(96,174,110,188)(97,173,111,187)(98,172,112,186)>;
G:=Group( (1,40,98,154)(2,41,99,155)(3,42,100,156)(4,43,101,157)(5,44,102,158)(6,45,103,159)(7,46,104,160)(8,47,105,161)(9,48,106,162)(10,49,107,163)(11,50,108,164)(12,51,109,165)(13,52,110,166)(14,53,111,167)(15,54,112,168)(16,55,85,141)(17,56,86,142)(18,29,87,143)(19,30,88,144)(20,31,89,145)(21,32,90,146)(22,33,91,147)(23,34,92,148)(24,35,93,149)(25,36,94,150)(26,37,95,151)(27,38,96,152)(28,39,97,153)(57,213,137,170)(58,214,138,171)(59,215,139,172)(60,216,140,173)(61,217,113,174)(62,218,114,175)(63,219,115,176)(64,220,116,177)(65,221,117,178)(66,222,118,179)(67,223,119,180)(68,224,120,181)(69,197,121,182)(70,198,122,183)(71,199,123,184)(72,200,124,185)(73,201,125,186)(74,202,126,187)(75,203,127,188)(76,204,128,189)(77,205,129,190)(78,206,130,191)(79,207,131,192)(80,208,132,193)(81,209,133,194)(82,210,134,195)(83,211,135,196)(84,212,136,169), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,215,15,201)(2,214,16,200)(3,213,17,199)(4,212,18,198)(5,211,19,197)(6,210,20,224)(7,209,21,223)(8,208,22,222)(9,207,23,221)(10,206,24,220)(11,205,25,219)(12,204,26,218)(13,203,27,217)(14,202,28,216)(29,122,43,136)(30,121,44,135)(31,120,45,134)(32,119,46,133)(33,118,47,132)(34,117,48,131)(35,116,49,130)(36,115,50,129)(37,114,51,128)(38,113,52,127)(39,140,53,126)(40,139,54,125)(41,138,55,124)(42,137,56,123)(57,142,71,156)(58,141,72,155)(59,168,73,154)(60,167,74,153)(61,166,75,152)(62,165,76,151)(63,164,77,150)(64,163,78,149)(65,162,79,148)(66,161,80,147)(67,160,81,146)(68,159,82,145)(69,158,83,144)(70,157,84,143)(85,185,99,171)(86,184,100,170)(87,183,101,169)(88,182,102,196)(89,181,103,195)(90,180,104,194)(91,179,105,193)(92,178,106,192)(93,177,107,191)(94,176,108,190)(95,175,109,189)(96,174,110,188)(97,173,111,187)(98,172,112,186) );
G=PermutationGroup([[(1,40,98,154),(2,41,99,155),(3,42,100,156),(4,43,101,157),(5,44,102,158),(6,45,103,159),(7,46,104,160),(8,47,105,161),(9,48,106,162),(10,49,107,163),(11,50,108,164),(12,51,109,165),(13,52,110,166),(14,53,111,167),(15,54,112,168),(16,55,85,141),(17,56,86,142),(18,29,87,143),(19,30,88,144),(20,31,89,145),(21,32,90,146),(22,33,91,147),(23,34,92,148),(24,35,93,149),(25,36,94,150),(26,37,95,151),(27,38,96,152),(28,39,97,153),(57,213,137,170),(58,214,138,171),(59,215,139,172),(60,216,140,173),(61,217,113,174),(62,218,114,175),(63,219,115,176),(64,220,116,177),(65,221,117,178),(66,222,118,179),(67,223,119,180),(68,224,120,181),(69,197,121,182),(70,198,122,183),(71,199,123,184),(72,200,124,185),(73,201,125,186),(74,202,126,187),(75,203,127,188),(76,204,128,189),(77,205,129,190),(78,206,130,191),(79,207,131,192),(80,208,132,193),(81,209,133,194),(82,210,134,195),(83,211,135,196),(84,212,136,169)], [(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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196),(197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,215,15,201),(2,214,16,200),(3,213,17,199),(4,212,18,198),(5,211,19,197),(6,210,20,224),(7,209,21,223),(8,208,22,222),(9,207,23,221),(10,206,24,220),(11,205,25,219),(12,204,26,218),(13,203,27,217),(14,202,28,216),(29,122,43,136),(30,121,44,135),(31,120,45,134),(32,119,46,133),(33,118,47,132),(34,117,48,131),(35,116,49,130),(36,115,50,129),(37,114,51,128),(38,113,52,127),(39,140,53,126),(40,139,54,125),(41,138,55,124),(42,137,56,123),(57,142,71,156),(58,141,72,155),(59,168,73,154),(60,167,74,153),(61,166,75,152),(62,165,76,151),(63,164,77,150),(64,163,78,149),(65,162,79,148),(66,161,80,147),(67,160,81,146),(68,159,82,145),(69,158,83,144),(70,157,84,143),(85,185,99,171),(86,184,100,170),(87,183,101,169),(88,182,102,196),(89,181,103,195),(90,180,104,194),(91,179,105,193),(92,178,106,192),(93,177,107,191),(94,176,108,190),(95,175,109,189),(96,174,110,188),(97,173,111,187),(98,172,112,186)]])
C4×Dic14 is a maximal subgroup of
C4.8Dic28 Dic14⋊2C8 C56⋊11Q8 C56⋊Q8 C42.16D14 Dic28⋊C4 C42.27D14 Dic14.3Q8 Dic14⋊C8 C28.M4(2) C42.36D14 Dic14⋊8D4 C4⋊Dic28 C28.7Q16 Dic14⋊4Q8 C42.51D14 C42.59D14 C42.61D14 Dic14.4Q8 Dic14⋊9D4 C28⋊Q16 Dic14⋊5Q8 Dic14⋊6Q8 C42.274D14 C42.277D14 C42.87D14 C42.88D14 C42.89D14 C42.91D14 C42.93D14 C42.96D14 C42.98D14 C42.99D14 C42.102D14 D4⋊5Dic14 C42.105D14 C42.106D14 D4⋊6Dic14 C42.108D14 Dic14⋊23D4 Dic14⋊24D4 C42.229D14 C42.114D14 C42.115D14 Dic14⋊10Q8 C42.122D14 Q8⋊5Dic14 Q8⋊6Dic14 C4×Q8×D7 C42.125D14 C42.232D14 C42.134D14 C42.135D14 C42.136D14 C42.137D14 C42.139D14 Dic14⋊10D4 C42.143D14 Dic14⋊7Q8 D28⋊7Q8 C42.152D14 C42.154D14 C42.159D14 C42.160D14 C42.162D14 C42.164D14 C42.166D14 Dic14⋊11D4 Dic14⋊8Q8 Dic14⋊9Q8 D28⋊8Q8 D28⋊9Q8 C42.177D14
C4×Dic14 is a maximal quotient of
(C2×C28)⋊Q8 C14.(C4×Q8) C4⋊Dic7⋊8C4 C14.(C4×D4) C56⋊11Q8 C56⋊Q8 C28⋊4(C4⋊C4) (C2×C28)⋊10Q8 (C2×C42).D7
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | Q8 | D7 | C4○D4 | D14 | Dic14 | C4×D7 | C4○D28 |
| kernel | C4×Dic14 | C4×Dic7 | Dic7⋊C4 | C4⋊Dic7 | C4×C28 | C2×Dic14 | Dic14 | C28 | C42 | C14 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 9 | 12 | 12 | 12 |
Matrix representation of C4×Dic14 ►in GL4(𝔽29) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 18 | 1 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
| 16 | 5 | 0 | 0 |
| 7 | 13 | 0 | 0 |
| 0 | 0 | 15 | 3 |
| 0 | 0 | 2 | 14 |
G:=sub<GL(4,GF(29))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[18,28,0,0,1,0,0,0,0,0,28,28,0,0,2,1],[16,7,0,0,5,13,0,0,0,0,15,2,0,0,3,14] >;
C4×Dic14 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_{14} % in TeX
G:=Group("C4xDic14"); // GroupNames label
G:=SmallGroup(224,63);
// by ID
G=gap.SmallGroup(224,63);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,50,6917]);
// Polycyclic
G:=Group<a,b,c|a^4=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations