metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊3D8, C42.218D14, C7⋊C8⋊11D4, C4⋊1(D4⋊D7), C4⋊1D4⋊2D7, C7⋊2(C8⋊4D4), C4.14(D4×D7), C14.58(C2×D8), C28.31(C2×D4), C28⋊4D4⋊10C2, (C2×D4).56D14, (C2×C28).148D4, C14.20(C4⋊1D4), C2.11(C28⋊D4), (C4×C28).121C22, (C2×C28).391C23, (D4×C14).72C22, (C2×D28).105C22, (C4×C7⋊C8)⋊15C2, (C2×D4⋊D7)⋊14C2, (C7×C4⋊1D4)⋊2C2, C2.13(C2×D4⋊D7), (C2×C14).522(C2×D4), (C2×C7⋊C8).257C22, (C2×C4).131(C7⋊D4), (C2×C4).489(C22×D7), C22.195(C2×C7⋊D4), SmallGroup(448,607)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊D8
G = < a,b,c | a28=b8=c2=1, bab-1=a13, cac=a-1, cbc=b-1 >
Subgroups: 1004 in 162 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C14, C42, C2×C8, D8, C2×D4, C2×D4, C28, D14, C2×C14, C2×C14, C4×C8, C4⋊1D4, C4⋊1D4, C2×D8, C7⋊C8, D28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊4D4, C2×C7⋊C8, D4⋊D7, C4×C28, C2×D28, C2×D28, D4×C14, D4×C14, C4×C7⋊C8, C28⋊4D4, C2×D4⋊D7, C7×C4⋊1D4, C28⋊D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C4⋊1D4, C2×D8, C7⋊D4, C22×D7, C8⋊4D4, D4⋊D7, D4×D7, C2×C7⋊D4, C2×D4⋊D7, C28⋊D4, C28⋊D8
►On 224 points
Generators in S224
(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)
(1 221 124 142 180 110 43 74)(2 206 125 155 181 95 44 59)(3 219 126 168 182 108 45 72)(4 204 127 153 183 93 46 57)(5 217 128 166 184 106 47 70)(6 202 129 151 185 91 48 83)(7 215 130 164 186 104 49 68)(8 200 131 149 187 89 50 81)(9 213 132 162 188 102 51 66)(10 198 133 147 189 87 52 79)(11 211 134 160 190 100 53 64)(12 224 135 145 191 85 54 77)(13 209 136 158 192 98 55 62)(14 222 137 143 193 111 56 75)(15 207 138 156 194 96 29 60)(16 220 139 141 195 109 30 73)(17 205 140 154 196 94 31 58)(18 218 113 167 169 107 32 71)(19 203 114 152 170 92 33 84)(20 216 115 165 171 105 34 69)(21 201 116 150 172 90 35 82)(22 214 117 163 173 103 36 67)(23 199 118 148 174 88 37 80)(24 212 119 161 175 101 38 65)(25 197 120 146 176 86 39 78)(26 210 121 159 177 99 40 63)(27 223 122 144 178 112 41 76)(28 208 123 157 179 97 42 61)
(1 187)(2 186)(3 185)(4 184)(5 183)(6 182)(7 181)(8 180)(9 179)(10 178)(11 177)(12 176)(13 175)(14 174)(15 173)(16 172)(17 171)(18 170)(19 169)(20 196)(21 195)(22 194)(23 193)(24 192)(25 191)(26 190)(27 189)(28 188)(29 36)(30 35)(31 34)(32 33)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(57 106)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(113 114)(115 140)(116 139)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(141 201)(142 200)(143 199)(144 198)(145 197)(146 224)(147 223)(148 222)(149 221)(150 220)(151 219)(152 218)(153 217)(154 216)(155 215)(156 214)(157 213)(158 212)(159 211)(160 210)(161 209)(162 208)(163 207)(164 206)(165 205)(166 204)(167 203)(168 202)
G:=sub<Sym(224)| (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), (1,221,124,142,180,110,43,74)(2,206,125,155,181,95,44,59)(3,219,126,168,182,108,45,72)(4,204,127,153,183,93,46,57)(5,217,128,166,184,106,47,70)(6,202,129,151,185,91,48,83)(7,215,130,164,186,104,49,68)(8,200,131,149,187,89,50,81)(9,213,132,162,188,102,51,66)(10,198,133,147,189,87,52,79)(11,211,134,160,190,100,53,64)(12,224,135,145,191,85,54,77)(13,209,136,158,192,98,55,62)(14,222,137,143,193,111,56,75)(15,207,138,156,194,96,29,60)(16,220,139,141,195,109,30,73)(17,205,140,154,196,94,31,58)(18,218,113,167,169,107,32,71)(19,203,114,152,170,92,33,84)(20,216,115,165,171,105,34,69)(21,201,116,150,172,90,35,82)(22,214,117,163,173,103,36,67)(23,199,118,148,174,88,37,80)(24,212,119,161,175,101,38,65)(25,197,120,146,176,86,39,78)(26,210,121,159,177,99,40,63)(27,223,122,144,178,112,41,76)(28,208,123,157,179,97,42,61), (1,187)(2,186)(3,185)(4,184)(5,183)(6,182)(7,181)(8,180)(9,179)(10,178)(11,177)(12,176)(13,175)(14,174)(15,173)(16,172)(17,171)(18,170)(19,169)(20,196)(21,195)(22,194)(23,193)(24,192)(25,191)(26,190)(27,189)(28,188)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(113,114)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(141,201)(142,200)(143,199)(144,198)(145,197)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218)(153,217)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)>;
G:=Group( (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), (1,221,124,142,180,110,43,74)(2,206,125,155,181,95,44,59)(3,219,126,168,182,108,45,72)(4,204,127,153,183,93,46,57)(5,217,128,166,184,106,47,70)(6,202,129,151,185,91,48,83)(7,215,130,164,186,104,49,68)(8,200,131,149,187,89,50,81)(9,213,132,162,188,102,51,66)(10,198,133,147,189,87,52,79)(11,211,134,160,190,100,53,64)(12,224,135,145,191,85,54,77)(13,209,136,158,192,98,55,62)(14,222,137,143,193,111,56,75)(15,207,138,156,194,96,29,60)(16,220,139,141,195,109,30,73)(17,205,140,154,196,94,31,58)(18,218,113,167,169,107,32,71)(19,203,114,152,170,92,33,84)(20,216,115,165,171,105,34,69)(21,201,116,150,172,90,35,82)(22,214,117,163,173,103,36,67)(23,199,118,148,174,88,37,80)(24,212,119,161,175,101,38,65)(25,197,120,146,176,86,39,78)(26,210,121,159,177,99,40,63)(27,223,122,144,178,112,41,76)(28,208,123,157,179,97,42,61), (1,187)(2,186)(3,185)(4,184)(5,183)(6,182)(7,181)(8,180)(9,179)(10,178)(11,177)(12,176)(13,175)(14,174)(15,173)(16,172)(17,171)(18,170)(19,169)(20,196)(21,195)(22,194)(23,193)(24,192)(25,191)(26,190)(27,189)(28,188)(29,36)(30,35)(31,34)(32,33)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(113,114)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,132)(124,131)(125,130)(126,129)(127,128)(141,201)(142,200)(143,199)(144,198)(145,197)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218)(153,217)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202) );
G=PermutationGroup([[(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)], [(1,221,124,142,180,110,43,74),(2,206,125,155,181,95,44,59),(3,219,126,168,182,108,45,72),(4,204,127,153,183,93,46,57),(5,217,128,166,184,106,47,70),(6,202,129,151,185,91,48,83),(7,215,130,164,186,104,49,68),(8,200,131,149,187,89,50,81),(9,213,132,162,188,102,51,66),(10,198,133,147,189,87,52,79),(11,211,134,160,190,100,53,64),(12,224,135,145,191,85,54,77),(13,209,136,158,192,98,55,62),(14,222,137,143,193,111,56,75),(15,207,138,156,194,96,29,60),(16,220,139,141,195,109,30,73),(17,205,140,154,196,94,31,58),(18,218,113,167,169,107,32,71),(19,203,114,152,170,92,33,84),(20,216,115,165,171,105,34,69),(21,201,116,150,172,90,35,82),(22,214,117,163,173,103,36,67),(23,199,118,148,174,88,37,80),(24,212,119,161,175,101,38,65),(25,197,120,146,176,86,39,78),(26,210,121,159,177,99,40,63),(27,223,122,144,178,112,41,76),(28,208,123,157,179,97,42,61)], [(1,187),(2,186),(3,185),(4,184),(5,183),(6,182),(7,181),(8,180),(9,179),(10,178),(11,177),(12,176),(13,175),(14,174),(15,173),(16,172),(17,171),(18,170),(19,169),(20,196),(21,195),(22,194),(23,193),(24,192),(25,191),(26,190),(27,189),(28,188),(29,36),(30,35),(31,34),(32,33),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(57,106),(58,105),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(113,114),(115,140),(116,139),(117,138),(118,137),(119,136),(120,135),(121,134),(122,133),(123,132),(124,131),(125,130),(126,129),(127,128),(141,201),(142,200),(143,199),(144,198),(145,197),(146,224),(147,223),(148,222),(149,221),(150,220),(151,219),(152,218),(153,217),(154,216),(155,215),(156,214),(157,213),(158,212),(159,211),(160,210),(161,209),(162,208),(163,207),(164,206),(165,205),(166,204),(167,203),(168,202)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 56 | 56 | 2 | ··· | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D14 | D14 | C7⋊D4 | D4⋊D7 | D4×D7 |
| kernel | C28⋊D8 | C4×C7⋊C8 | C28⋊4D4 | C2×D4⋊D7 | C7×C4⋊1D4 | C7⋊C8 | C2×C28 | C4⋊1D4 | C28 | C42 | C2×D4 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 3 | 8 | 3 | 6 | 12 | 12 | 6 |
Matrix representation of C28⋊D8 ►in GL6(𝔽113)
| 89 | 34 | 0 | 0 | 0 | 0 |
| 103 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 111 | 0 | 0 |
| 0 | 0 | 53 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 2 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 106 | 31 | 0 | 0 | 0 | 0 |
| 64 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 52 | 2 | 0 | 0 |
| 0 | 0 | 60 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 62 |
| 0 | 0 | 0 | 0 | 82 | 62 |
| 0 | 79 | 0 | 0 | 0 | 0 |
| 103 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 111 | 0 | 0 |
| 0 | 0 | 52 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [89,103,0,0,0,0,34,0,0,0,0,0,0,0,61,53,0,0,0,0,111,52,0,0,0,0,0,0,112,112,0,0,0,0,2,1],[106,64,0,0,0,0,31,7,0,0,0,0,0,0,52,60,0,0,0,0,2,61,0,0,0,0,0,0,0,82,0,0,0,0,62,62],[0,103,0,0,0,0,79,0,0,0,0,0,0,0,61,52,0,0,0,0,111,52,0,0,0,0,0,0,1,0,0,0,0,0,111,112] >;
C28⋊D8 in GAP, Magma, Sage, TeX
C_{28}\rtimes D_8 % in TeX
G:=Group("C28:D8"); // GroupNames label
G:=SmallGroup(448,607);
// by ID
G=gap.SmallGroup(448,607);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,219,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations