metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊7D28, C56⋊4D4, D14⋊2D8, C2.D8⋊3D7, C7⋊2(C8⋊7D4), C2.14(D7×D8), (C2×D56)⋊16C2, C4⋊D28⋊7C2, C4⋊C4.48D14, C14.30(C2×D8), C4.53(C2×D28), C28.133(C2×D4), (C2×C8).231D14, C14.D8⋊21C2, C28.38(C4○D4), C14.75(C4○D8), (C2×C56).83C22, C4.9(Q8⋊2D7), (C22×D7).55D4, C22.229(D4×D7), C14.46(C4⋊D4), C2.19(C4⋊D28), (C2×C28).299C23, (C2×Dic7).103D4, (C2×D28).82C22, C2.13(Q8.D14), (D7×C2×C8)⋊2C2, (C7×C2.D8)⋊5C2, (C2×C14).304(C2×D4), (C7×C4⋊C4).92C22, (C2×C7⋊C8).235C22, (C2×C4×D7).236C22, (C2×C4).402(C22×D7), SmallGroup(448,417)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊7D28
G = < a,b,c | a8=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 972 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C8⋊7D4, C8×D7, D56, C2×C7⋊C8, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, C14.D8, C7×C2.D8, C4⋊D28, D7×C2×C8, C2×D56, C8⋊7D28
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, C4○D4, D14, C4⋊D4, C2×D8, C4○D8, D28, C22×D7, C8⋊7D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, D7×D8, Q8.D14, C8⋊7D28
►On 224 points
Generators in S224
(1 184 55 131 77 167 91 208)(2 209 92 168 78 132 56 185)(3 186 29 133 79 141 93 210)(4 211 94 142 80 134 30 187)(5 188 31 135 81 143 95 212)(6 213 96 144 82 136 32 189)(7 190 33 137 83 145 97 214)(8 215 98 146 84 138 34 191)(9 192 35 139 57 147 99 216)(10 217 100 148 58 140 36 193)(11 194 37 113 59 149 101 218)(12 219 102 150 60 114 38 195)(13 196 39 115 61 151 103 220)(14 221 104 152 62 116 40 169)(15 170 41 117 63 153 105 222)(16 223 106 154 64 118 42 171)(17 172 43 119 65 155 107 224)(18 197 108 156 66 120 44 173)(19 174 45 121 67 157 109 198)(20 199 110 158 68 122 46 175)(21 176 47 123 69 159 111 200)(22 201 112 160 70 124 48 177)(23 178 49 125 71 161 85 202)(24 203 86 162 72 126 50 179)(25 180 51 127 73 163 87 204)(26 205 88 164 74 128 52 181)(27 182 53 129 75 165 89 206)(28 207 90 166 76 130 54 183)
(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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 88)(30 87)(31 86)(32 85)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(41 104)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(113 197)(114 224)(115 223)(116 222)(117 221)(118 220)(119 219)(120 218)(121 217)(122 216)(123 215)(124 214)(125 213)(126 212)(127 211)(128 210)(129 209)(130 208)(131 207)(132 206)(133 205)(134 204)(135 203)(136 202)(137 201)(138 200)(139 199)(140 198)(141 181)(142 180)(143 179)(144 178)(145 177)(146 176)(147 175)(148 174)(149 173)(150 172)(151 171)(152 170)(153 169)(154 196)(155 195)(156 194)(157 193)(158 192)(159 191)(160 190)(161 189)(162 188)(163 187)(164 186)(165 185)(166 184)(167 183)(168 182)
G:=sub<Sym(224)| (1,184,55,131,77,167,91,208)(2,209,92,168,78,132,56,185)(3,186,29,133,79,141,93,210)(4,211,94,142,80,134,30,187)(5,188,31,135,81,143,95,212)(6,213,96,144,82,136,32,189)(7,190,33,137,83,145,97,214)(8,215,98,146,84,138,34,191)(9,192,35,139,57,147,99,216)(10,217,100,148,58,140,36,193)(11,194,37,113,59,149,101,218)(12,219,102,150,60,114,38,195)(13,196,39,115,61,151,103,220)(14,221,104,152,62,116,40,169)(15,170,41,117,63,153,105,222)(16,223,106,154,64,118,42,171)(17,172,43,119,65,155,107,224)(18,197,108,156,66,120,44,173)(19,174,45,121,67,157,109,198)(20,199,110,158,68,122,46,175)(21,176,47,123,69,159,111,200)(22,201,112,160,70,124,48,177)(23,178,49,125,71,161,85,202)(24,203,86,162,72,126,50,179)(25,180,51,127,73,163,87,204)(26,205,88,164,74,128,52,181)(27,182,53,129,75,165,89,206)(28,207,90,166,76,130,54,183), (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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,88)(30,87)(31,86)(32,85)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(113,197)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)(120,218)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,200)(139,199)(140,198)(141,181)(142,180)(143,179)(144,178)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,196)(155,195)(156,194)(157,193)(158,192)(159,191)(160,190)(161,189)(162,188)(163,187)(164,186)(165,185)(166,184)(167,183)(168,182)>;
G:=Group( (1,184,55,131,77,167,91,208)(2,209,92,168,78,132,56,185)(3,186,29,133,79,141,93,210)(4,211,94,142,80,134,30,187)(5,188,31,135,81,143,95,212)(6,213,96,144,82,136,32,189)(7,190,33,137,83,145,97,214)(8,215,98,146,84,138,34,191)(9,192,35,139,57,147,99,216)(10,217,100,148,58,140,36,193)(11,194,37,113,59,149,101,218)(12,219,102,150,60,114,38,195)(13,196,39,115,61,151,103,220)(14,221,104,152,62,116,40,169)(15,170,41,117,63,153,105,222)(16,223,106,154,64,118,42,171)(17,172,43,119,65,155,107,224)(18,197,108,156,66,120,44,173)(19,174,45,121,67,157,109,198)(20,199,110,158,68,122,46,175)(21,176,47,123,69,159,111,200)(22,201,112,160,70,124,48,177)(23,178,49,125,71,161,85,202)(24,203,86,162,72,126,50,179)(25,180,51,127,73,163,87,204)(26,205,88,164,74,128,52,181)(27,182,53,129,75,165,89,206)(28,207,90,166,76,130,54,183), (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), (1,76)(2,75)(3,74)(4,73)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,88)(30,87)(31,86)(32,85)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(113,197)(114,224)(115,223)(116,222)(117,221)(118,220)(119,219)(120,218)(121,217)(122,216)(123,215)(124,214)(125,213)(126,212)(127,211)(128,210)(129,209)(130,208)(131,207)(132,206)(133,205)(134,204)(135,203)(136,202)(137,201)(138,200)(139,199)(140,198)(141,181)(142,180)(143,179)(144,178)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,196)(155,195)(156,194)(157,193)(158,192)(159,191)(160,190)(161,189)(162,188)(163,187)(164,186)(165,185)(166,184)(167,183)(168,182) );
G=PermutationGroup([[(1,184,55,131,77,167,91,208),(2,209,92,168,78,132,56,185),(3,186,29,133,79,141,93,210),(4,211,94,142,80,134,30,187),(5,188,31,135,81,143,95,212),(6,213,96,144,82,136,32,189),(7,190,33,137,83,145,97,214),(8,215,98,146,84,138,34,191),(9,192,35,139,57,147,99,216),(10,217,100,148,58,140,36,193),(11,194,37,113,59,149,101,218),(12,219,102,150,60,114,38,195),(13,196,39,115,61,151,103,220),(14,221,104,152,62,116,40,169),(15,170,41,117,63,153,105,222),(16,223,106,154,64,118,42,171),(17,172,43,119,65,155,107,224),(18,197,108,156,66,120,44,173),(19,174,45,121,67,157,109,198),(20,199,110,158,68,122,46,175),(21,176,47,123,69,159,111,200),(22,201,112,160,70,124,48,177),(23,178,49,125,71,161,85,202),(24,203,86,162,72,126,50,179),(25,180,51,127,73,163,87,204),(26,205,88,164,74,128,52,181),(27,182,53,129,75,165,89,206),(28,207,90,166,76,130,54,183)], [(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)], [(1,76),(2,75),(3,74),(4,73),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,84),(22,83),(23,82),(24,81),(25,80),(26,79),(27,78),(28,77),(29,88),(30,87),(31,86),(32,85),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(41,104),(42,103),(43,102),(44,101),(45,100),(46,99),(47,98),(48,97),(49,96),(50,95),(51,94),(52,93),(53,92),(54,91),(55,90),(56,89),(113,197),(114,224),(115,223),(116,222),(117,221),(118,220),(119,219),(120,218),(121,217),(122,216),(123,215),(124,214),(125,213),(126,212),(127,211),(128,210),(129,209),(130,208),(131,207),(132,206),(133,205),(134,204),(135,203),(136,202),(137,201),(138,200),(139,199),(140,198),(141,181),(142,180),(143,179),(144,178),(145,177),(146,176),(147,175),(148,174),(149,173),(150,172),(151,171),(152,170),(153,169),(154,196),(155,195),(156,194),(157,193),(158,192),(159,191),(160,190),(161,189),(162,188),(163,187),(164,186),(165,185),(166,184),(167,183),(168,182)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 56 | 56 | 2 | 2 | 8 | 8 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D8 | D14 | D14 | C4○D8 | D28 | Q8⋊2D7 | D4×D7 | D7×D8 | Q8.D14 |
| kernel | C8⋊7D28 | C14.D8 | C7×C2.D8 | C4⋊D28 | D7×C2×C8 | C2×D56 | C56 | C2×Dic7 | C22×D7 | C2.D8 | C28 | D14 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C8⋊7D28 ►in GL6(𝔽113)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 93 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 95 | 3 | 0 | 0 |
| 0 | 0 | 0 | 69 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 89 | 36 | 0 | 0 | 0 | 0 |
| 75 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 86 | 0 | 0 |
| 0 | 0 | 77 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 89 |
| 0 | 0 | 0 | 0 | 24 | 10 |
| 89 | 36 | 0 | 0 | 0 | 0 |
| 6 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 65 | 42 | 0 | 0 |
| 0 | 0 | 77 | 48 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
G:=sub<GL(6,GF(113))| [98,93,0,0,0,0,0,15,0,0,0,0,0,0,95,0,0,0,0,0,3,69,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[89,75,0,0,0,0,36,24,0,0,0,0,0,0,65,77,0,0,0,0,86,48,0,0,0,0,0,0,112,24,0,0,0,0,89,10],[89,6,0,0,0,0,36,24,0,0,0,0,0,0,65,77,0,0,0,0,42,48,0,0,0,0,0,0,112,24,0,0,0,0,0,1] >;
C8⋊7D28 in GAP, Magma, Sage, TeX
C_8\rtimes_7D_{28} % in TeX
G:=Group("C8:7D28"); // GroupNames label
G:=SmallGroup(448,417);
// by ID
G=gap.SmallGroup(448,417);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations