metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6D28, C56⋊25D4, C28⋊4M4(2), C42.257D14, (C4×C8)⋊14D7, (C4×C56)⋊21C2, D14⋊C8⋊1C2, C7⋊1(C8⋊6D4), C28⋊C8⋊2C2, C14.5(C4×D4), C2.8(C4×D28), C4⋊2(C8⋊D7), D14⋊C4.1C4, (C4×D28).2C2, (C2×D28).6C4, C4.75(C2×D28), C4⋊Dic7.7C4, C14.4(C8○D4), (C2×C8).283D14, C28.295(C2×D4), C14.4(C2×M4(2)), C4.126(C4○D28), C28.242(C4○D4), (C4×C28).324C22, (C2×C28).806C23, (C2×C56).343C22, C2.7(D28.2C4), C2.7(C2×C8⋊D7), (C2×C8⋊D7)⋊10C2, C22.95(C2×C4×D7), (C2×C4).104(C4×D7), (C2×C28).221(C2×C4), (C2×C7⋊C8).187C22, (C2×C4×D7).175C22, (C22×D7).9(C2×C4), (C2×C14).61(C22×C4), (C2×Dic7).11(C2×C4), (C2×C4).748(C22×D7), SmallGroup(448,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊6D28
G = < a,b,c | a8=b28=c2=1, ab=ba, cac=a5, cbc=b-1 >
Subgroups: 580 in 122 conjugacy classes, 55 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, Dic7, C28, C28, C28, D14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C7⋊C8, C56, C56, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C8⋊6D4, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C28⋊C8, D14⋊C8, C4×C56, C4×D28, C2×C8⋊D7, C8⋊6D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, D28, C22×D7, C8⋊6D4, C8⋊D7, C2×C4×D7, C2×D28, C4○D28, C4×D28, C2×C8⋊D7, D28.2C4, C8⋊6D28
►On 224 points
Generators in S224
(1 31 178 131 58 160 112 208)(2 32 179 132 59 161 85 209)(3 33 180 133 60 162 86 210)(4 34 181 134 61 163 87 211)(5 35 182 135 62 164 88 212)(6 36 183 136 63 165 89 213)(7 37 184 137 64 166 90 214)(8 38 185 138 65 167 91 215)(9 39 186 139 66 168 92 216)(10 40 187 140 67 141 93 217)(11 41 188 113 68 142 94 218)(12 42 189 114 69 143 95 219)(13 43 190 115 70 144 96 220)(14 44 191 116 71 145 97 221)(15 45 192 117 72 146 98 222)(16 46 193 118 73 147 99 223)(17 47 194 119 74 148 100 224)(18 48 195 120 75 149 101 197)(19 49 196 121 76 150 102 198)(20 50 169 122 77 151 103 199)(21 51 170 123 78 152 104 200)(22 52 171 124 79 153 105 201)(23 53 172 125 80 154 106 202)(24 54 173 126 81 155 107 203)(25 55 174 127 82 156 108 204)(26 56 175 128 83 157 109 205)(27 29 176 129 84 158 110 206)(28 30 177 130 57 159 111 207)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 168)(30 167)(31 166)(32 165)(33 164)(34 163)(35 162)(36 161)(37 160)(38 159)(39 158)(40 157)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 65)(58 64)(59 63)(60 62)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(85 89)(86 88)(90 112)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(113 204)(114 203)(115 202)(116 201)(117 200)(118 199)(119 198)(120 197)(121 224)(122 223)(123 222)(124 221)(125 220)(126 219)(127 218)(128 217)(129 216)(130 215)(131 214)(132 213)(133 212)(134 211)(135 210)(136 209)(137 208)(138 207)(139 206)(140 205)(169 193)(170 192)(171 191)(172 190)(173 189)(174 188)(175 187)(176 186)(177 185)(178 184)(179 183)(180 182)(194 196)
G:=sub<Sym(224)| (1,31,178,131,58,160,112,208)(2,32,179,132,59,161,85,209)(3,33,180,133,60,162,86,210)(4,34,181,134,61,163,87,211)(5,35,182,135,62,164,88,212)(6,36,183,136,63,165,89,213)(7,37,184,137,64,166,90,214)(8,38,185,138,65,167,91,215)(9,39,186,139,66,168,92,216)(10,40,187,140,67,141,93,217)(11,41,188,113,68,142,94,218)(12,42,189,114,69,143,95,219)(13,43,190,115,70,144,96,220)(14,44,191,116,71,145,97,221)(15,45,192,117,72,146,98,222)(16,46,193,118,73,147,99,223)(17,47,194,119,74,148,100,224)(18,48,195,120,75,149,101,197)(19,49,196,121,76,150,102,198)(20,50,169,122,77,151,103,199)(21,51,170,123,78,152,104,200)(22,52,171,124,79,153,105,201)(23,53,172,125,80,154,106,202)(24,54,173,126,81,155,107,203)(25,55,174,127,82,156,108,204)(26,56,175,128,83,157,109,205)(27,29,176,129,84,158,110,206)(28,30,177,130,57,159,111,207), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(113,204)(114,203)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)(128,217)(129,216)(130,215)(131,214)(132,213)(133,212)(134,211)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(169,193)(170,192)(171,191)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(194,196)>;
G:=Group( (1,31,178,131,58,160,112,208)(2,32,179,132,59,161,85,209)(3,33,180,133,60,162,86,210)(4,34,181,134,61,163,87,211)(5,35,182,135,62,164,88,212)(6,36,183,136,63,165,89,213)(7,37,184,137,64,166,90,214)(8,38,185,138,65,167,91,215)(9,39,186,139,66,168,92,216)(10,40,187,140,67,141,93,217)(11,41,188,113,68,142,94,218)(12,42,189,114,69,143,95,219)(13,43,190,115,70,144,96,220)(14,44,191,116,71,145,97,221)(15,45,192,117,72,146,98,222)(16,46,193,118,73,147,99,223)(17,47,194,119,74,148,100,224)(18,48,195,120,75,149,101,197)(19,49,196,121,76,150,102,198)(20,50,169,122,77,151,103,199)(21,51,170,123,78,152,104,200)(22,52,171,124,79,153,105,201)(23,53,172,125,80,154,106,202)(24,54,173,126,81,155,107,203)(25,55,174,127,82,156,108,204)(26,56,175,128,83,157,109,205)(27,29,176,129,84,158,110,206)(28,30,177,130,57,159,111,207), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,65)(58,64)(59,63)(60,62)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(85,89)(86,88)(90,112)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(113,204)(114,203)(115,202)(116,201)(117,200)(118,199)(119,198)(120,197)(121,224)(122,223)(123,222)(124,221)(125,220)(126,219)(127,218)(128,217)(129,216)(130,215)(131,214)(132,213)(133,212)(134,211)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(169,193)(170,192)(171,191)(172,190)(173,189)(174,188)(175,187)(176,186)(177,185)(178,184)(179,183)(180,182)(194,196) );
G=PermutationGroup([[(1,31,178,131,58,160,112,208),(2,32,179,132,59,161,85,209),(3,33,180,133,60,162,86,210),(4,34,181,134,61,163,87,211),(5,35,182,135,62,164,88,212),(6,36,183,136,63,165,89,213),(7,37,184,137,64,166,90,214),(8,38,185,138,65,167,91,215),(9,39,186,139,66,168,92,216),(10,40,187,140,67,141,93,217),(11,41,188,113,68,142,94,218),(12,42,189,114,69,143,95,219),(13,43,190,115,70,144,96,220),(14,44,191,116,71,145,97,221),(15,45,192,117,72,146,98,222),(16,46,193,118,73,147,99,223),(17,47,194,119,74,148,100,224),(18,48,195,120,75,149,101,197),(19,49,196,121,76,150,102,198),(20,50,169,122,77,151,103,199),(21,51,170,123,78,152,104,200),(22,52,171,124,79,153,105,201),(23,53,172,125,80,154,106,202),(24,54,173,126,81,155,107,203),(25,55,174,127,82,156,108,204),(26,56,175,128,83,157,109,205),(27,29,176,129,84,158,110,206),(28,30,177,130,57,159,111,207)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,168),(30,167),(31,166),(32,165),(33,164),(34,163),(35,162),(36,161),(37,160),(38,159),(39,158),(40,157),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,65),(58,64),(59,63),(60,62),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(85,89),(86,88),(90,112),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(113,204),(114,203),(115,202),(116,201),(117,200),(118,199),(119,198),(120,197),(121,224),(122,223),(123,222),(124,221),(125,220),(126,219),(127,218),(128,217),(129,216),(130,215),(131,214),(132,213),(133,212),(134,211),(135,210),(136,209),(137,208),(138,207),(139,206),(140,205),(169,193),(170,192),(171,191),(172,190),(173,189),(174,188),(175,187),(176,186),(177,185),(178,184),(179,183),(180,182),(194,196)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | M4(2) | C4○D4 | D14 | D14 | C8○D4 | D28 | C4×D7 | C8⋊D7 | C4○D28 | D28.2C4 |
| kernel | C8⋊6D28 | C28⋊C8 | D14⋊C8 | C4×C56 | C4×D28 | C2×C8⋊D7 | C4⋊Dic7 | D14⋊C4 | C2×D28 | C56 | C4×C8 | C28 | C28 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 4 | 12 | 12 | 24 | 12 | 24 |
Matrix representation of C8⋊6D28 ►in GL4(𝔽113) generated by
| 66 | 23 | 0 | 0 |
| 90 | 47 | 0 | 0 |
| 0 | 0 | 96 | 87 |
| 0 | 0 | 3 | 17 |
| 33 | 9 | 0 | 0 |
| 104 | 1 | 0 | 0 |
| 0 | 0 | 33 | 97 |
| 0 | 0 | 54 | 80 |
| 80 | 104 | 0 | 0 |
| 33 | 33 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 10 | 1 |
G:=sub<GL(4,GF(113))| [66,90,0,0,23,47,0,0,0,0,96,3,0,0,87,17],[33,104,0,0,9,1,0,0,0,0,33,54,0,0,97,80],[80,33,0,0,104,33,0,0,0,0,112,10,0,0,0,1] >;
C8⋊6D28 in GAP, Magma, Sage, TeX
C_8\rtimes_6D_{28} % in TeX
G:=Group("C8:6D28"); // GroupNames label
G:=SmallGroup(448,222);
// by ID
G=gap.SmallGroup(448,222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,758,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;
// generators/relations