metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊2C4, C8⋊2Dic7, C28.4Q8, C14.2SD16, C4.4Dic14, C22.8D28, (C2×C8).6D7, C7⋊2(C4.Q8), (C2×C56).8C2, C14.5(C4⋊C4), C28.33(C2×C4), (C2×C14).13D4, (C2×C4).68D14, C4⋊Dic7.2C2, C4.6(C2×Dic7), C2.2(C56⋊C2), C2.3(C4⋊Dic7), (C2×C28).81C22, SmallGroup(224,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊Dic7
G = < a,b,c | a8=b14=1, c2=b7, ab=ba, cac-1=a3, cbc-1=b-1 >
Smallest permutation representation of C8⋊Dic7
►Regular action on 224 points
Generators in S224
(1 50 120 157 222 112 151 83)(2 51 121 158 223 99 152 84)(3 52 122 159 224 100 153 71)(4 53 123 160 211 101 154 72)(5 54 124 161 212 102 141 73)(6 55 125 162 213 103 142 74)(7 56 126 163 214 104 143 75)(8 43 113 164 215 105 144 76)(9 44 114 165 216 106 145 77)(10 45 115 166 217 107 146 78)(11 46 116 167 218 108 147 79)(12 47 117 168 219 109 148 80)(13 48 118 155 220 110 149 81)(14 49 119 156 221 111 150 82)(15 67 29 193 127 173 210 92)(16 68 30 194 128 174 197 93)(17 69 31 195 129 175 198 94)(18 70 32 196 130 176 199 95)(19 57 33 183 131 177 200 96)(20 58 34 184 132 178 201 97)(21 59 35 185 133 179 202 98)(22 60 36 186 134 180 203 85)(23 61 37 187 135 181 204 86)(24 62 38 188 136 182 205 87)(25 63 39 189 137 169 206 88)(26 64 40 190 138 170 207 89)(27 65 41 191 139 171 208 90)(28 66 42 192 140 172 209 91)
(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 66 8 59)(2 65 9 58)(3 64 10 57)(4 63 11 70)(5 62 12 69)(6 61 13 68)(7 60 14 67)(15 104 22 111)(16 103 23 110)(17 102 24 109)(18 101 25 108)(19 100 26 107)(20 99 27 106)(21 112 28 105)(29 163 36 156)(30 162 37 155)(31 161 38 168)(32 160 39 167)(33 159 40 166)(34 158 41 165)(35 157 42 164)(43 133 50 140)(44 132 51 139)(45 131 52 138)(46 130 53 137)(47 129 54 136)(48 128 55 135)(49 127 56 134)(71 207 78 200)(72 206 79 199)(73 205 80 198)(74 204 81 197)(75 203 82 210)(76 202 83 209)(77 201 84 208)(85 119 92 126)(86 118 93 125)(87 117 94 124)(88 116 95 123)(89 115 96 122)(90 114 97 121)(91 113 98 120)(141 188 148 195)(142 187 149 194)(143 186 150 193)(144 185 151 192)(145 184 152 191)(146 183 153 190)(147 196 154 189)(169 218 176 211)(170 217 177 224)(171 216 178 223)(172 215 179 222)(173 214 180 221)(174 213 181 220)(175 212 182 219)
G:=sub<Sym(224)| (1,50,120,157,222,112,151,83)(2,51,121,158,223,99,152,84)(3,52,122,159,224,100,153,71)(4,53,123,160,211,101,154,72)(5,54,124,161,212,102,141,73)(6,55,125,162,213,103,142,74)(7,56,126,163,214,104,143,75)(8,43,113,164,215,105,144,76)(9,44,114,165,216,106,145,77)(10,45,115,166,217,107,146,78)(11,46,116,167,218,108,147,79)(12,47,117,168,219,109,148,80)(13,48,118,155,220,110,149,81)(14,49,119,156,221,111,150,82)(15,67,29,193,127,173,210,92)(16,68,30,194,128,174,197,93)(17,69,31,195,129,175,198,94)(18,70,32,196,130,176,199,95)(19,57,33,183,131,177,200,96)(20,58,34,184,132,178,201,97)(21,59,35,185,133,179,202,98)(22,60,36,186,134,180,203,85)(23,61,37,187,135,181,204,86)(24,62,38,188,136,182,205,87)(25,63,39,189,137,169,206,88)(26,64,40,190,138,170,207,89)(27,65,41,191,139,171,208,90)(28,66,42,192,140,172,209,91), (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), (1,66,8,59)(2,65,9,58)(3,64,10,57)(4,63,11,70)(5,62,12,69)(6,61,13,68)(7,60,14,67)(15,104,22,111)(16,103,23,110)(17,102,24,109)(18,101,25,108)(19,100,26,107)(20,99,27,106)(21,112,28,105)(29,163,36,156)(30,162,37,155)(31,161,38,168)(32,160,39,167)(33,159,40,166)(34,158,41,165)(35,157,42,164)(43,133,50,140)(44,132,51,139)(45,131,52,138)(46,130,53,137)(47,129,54,136)(48,128,55,135)(49,127,56,134)(71,207,78,200)(72,206,79,199)(73,205,80,198)(74,204,81,197)(75,203,82,210)(76,202,83,209)(77,201,84,208)(85,119,92,126)(86,118,93,125)(87,117,94,124)(88,116,95,123)(89,115,96,122)(90,114,97,121)(91,113,98,120)(141,188,148,195)(142,187,149,194)(143,186,150,193)(144,185,151,192)(145,184,152,191)(146,183,153,190)(147,196,154,189)(169,218,176,211)(170,217,177,224)(171,216,178,223)(172,215,179,222)(173,214,180,221)(174,213,181,220)(175,212,182,219)>;
G:=Group( (1,50,120,157,222,112,151,83)(2,51,121,158,223,99,152,84)(3,52,122,159,224,100,153,71)(4,53,123,160,211,101,154,72)(5,54,124,161,212,102,141,73)(6,55,125,162,213,103,142,74)(7,56,126,163,214,104,143,75)(8,43,113,164,215,105,144,76)(9,44,114,165,216,106,145,77)(10,45,115,166,217,107,146,78)(11,46,116,167,218,108,147,79)(12,47,117,168,219,109,148,80)(13,48,118,155,220,110,149,81)(14,49,119,156,221,111,150,82)(15,67,29,193,127,173,210,92)(16,68,30,194,128,174,197,93)(17,69,31,195,129,175,198,94)(18,70,32,196,130,176,199,95)(19,57,33,183,131,177,200,96)(20,58,34,184,132,178,201,97)(21,59,35,185,133,179,202,98)(22,60,36,186,134,180,203,85)(23,61,37,187,135,181,204,86)(24,62,38,188,136,182,205,87)(25,63,39,189,137,169,206,88)(26,64,40,190,138,170,207,89)(27,65,41,191,139,171,208,90)(28,66,42,192,140,172,209,91), (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), (1,66,8,59)(2,65,9,58)(3,64,10,57)(4,63,11,70)(5,62,12,69)(6,61,13,68)(7,60,14,67)(15,104,22,111)(16,103,23,110)(17,102,24,109)(18,101,25,108)(19,100,26,107)(20,99,27,106)(21,112,28,105)(29,163,36,156)(30,162,37,155)(31,161,38,168)(32,160,39,167)(33,159,40,166)(34,158,41,165)(35,157,42,164)(43,133,50,140)(44,132,51,139)(45,131,52,138)(46,130,53,137)(47,129,54,136)(48,128,55,135)(49,127,56,134)(71,207,78,200)(72,206,79,199)(73,205,80,198)(74,204,81,197)(75,203,82,210)(76,202,83,209)(77,201,84,208)(85,119,92,126)(86,118,93,125)(87,117,94,124)(88,116,95,123)(89,115,96,122)(90,114,97,121)(91,113,98,120)(141,188,148,195)(142,187,149,194)(143,186,150,193)(144,185,151,192)(145,184,152,191)(146,183,153,190)(147,196,154,189)(169,218,176,211)(170,217,177,224)(171,216,178,223)(172,215,179,222)(173,214,180,221)(174,213,181,220)(175,212,182,219) );
G=PermutationGroup([[(1,50,120,157,222,112,151,83),(2,51,121,158,223,99,152,84),(3,52,122,159,224,100,153,71),(4,53,123,160,211,101,154,72),(5,54,124,161,212,102,141,73),(6,55,125,162,213,103,142,74),(7,56,126,163,214,104,143,75),(8,43,113,164,215,105,144,76),(9,44,114,165,216,106,145,77),(10,45,115,166,217,107,146,78),(11,46,116,167,218,108,147,79),(12,47,117,168,219,109,148,80),(13,48,118,155,220,110,149,81),(14,49,119,156,221,111,150,82),(15,67,29,193,127,173,210,92),(16,68,30,194,128,174,197,93),(17,69,31,195,129,175,198,94),(18,70,32,196,130,176,199,95),(19,57,33,183,131,177,200,96),(20,58,34,184,132,178,201,97),(21,59,35,185,133,179,202,98),(22,60,36,186,134,180,203,85),(23,61,37,187,135,181,204,86),(24,62,38,188,136,182,205,87),(25,63,39,189,137,169,206,88),(26,64,40,190,138,170,207,89),(27,65,41,191,139,171,208,90),(28,66,42,192,140,172,209,91)], [(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)], [(1,66,8,59),(2,65,9,58),(3,64,10,57),(4,63,11,70),(5,62,12,69),(6,61,13,68),(7,60,14,67),(15,104,22,111),(16,103,23,110),(17,102,24,109),(18,101,25,108),(19,100,26,107),(20,99,27,106),(21,112,28,105),(29,163,36,156),(30,162,37,155),(31,161,38,168),(32,160,39,167),(33,159,40,166),(34,158,41,165),(35,157,42,164),(43,133,50,140),(44,132,51,139),(45,131,52,138),(46,130,53,137),(47,129,54,136),(48,128,55,135),(49,127,56,134),(71,207,78,200),(72,206,79,199),(73,205,80,198),(74,204,81,197),(75,203,82,210),(76,202,83,209),(77,201,84,208),(85,119,92,126),(86,118,93,125),(87,117,94,124),(88,116,95,123),(89,115,96,122),(90,114,97,121),(91,113,98,120),(141,188,148,195),(142,187,149,194),(143,186,150,193),(144,185,151,192),(145,184,152,191),(146,183,153,190),(147,196,154,189),(169,218,176,211),(170,217,177,224),(171,216,178,223),(172,215,179,222),(173,214,180,221),(174,213,181,220),(175,212,182,219)]])
C8⋊Dic7 is a maximal subgroup of
C8.Dic14 C16⋊Dic7 D56⋊2C4 D8⋊2Dic7 C56⋊9Q8 C56.13Q8 C4×C56⋊C2 C8⋊Dic14 D56⋊C4 Dic28⋊C4 C23.34D28 C23.10D28 C23.38D28 C23.13D28 D4⋊Dic14 D4.Dic14 D14.SD16 C8⋊Dic7⋊C2 Q8⋊Dic14 Q8.2Dic14 D14.1SD16 (C2×C8).D14 Dic14.3Q8 D28⋊3Q8 D28.3Q8 Dic14⋊4Q8 C56⋊5Q8 C56.8Q8 D7×C4.Q8 (C8×D7)⋊C4 C56⋊4Q8 C56⋊(C2×C4) C23.22D28 C56⋊30D4 C23.47D28 C56⋊3D4 C56.4D4 D8⋊Dic7 C56⋊12D4 SD16×Dic7 C56⋊14D4 Q16⋊Dic7 C56.36D4
C8⋊Dic7 is a maximal quotient of
C56⋊2C8 C16⋊Dic7 C28.9C42
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D7 | SD16 | Dic7 | D14 | Dic14 | D28 | C56⋊C2 |
| kernel | C8⋊Dic7 | C4⋊Dic7 | C2×C56 | C56 | C28 | C2×C14 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 4 | 6 | 3 | 6 | 6 | 24 |
Matrix representation of C8⋊Dic7 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 69 | 0 | 0 |
| 0 | 0 | 0 | 9 | 8 |
| 0 | 0 | 0 | 18 | 104 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 87 | 25 |
| 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 49 |
| 0 | 0 | 0 | 4 | 101 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,18,0,0,0,0,0,69,0,0,0,0,0,9,18,0,0,0,8,104],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,87,0,0,0,1,25],[15,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,49,101] >;
C8⋊Dic7 in GAP, Magma, Sage, TeX
C_8\rtimes {\rm Dic}_7 % in TeX
G:=Group("C8:Dic7"); // GroupNames label
G:=SmallGroup(224,23);
// by ID
G=gap.SmallGroup(224,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,579,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^8=b^14=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
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