metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊8D28, C56⋊13D4, D14⋊3SD16, C4.Q8⋊8D7, C7⋊3(C8⋊8D4), C4⋊C4.38D14, C4.50(C2×D28), D14⋊2Q8⋊6C2, (C2×C8).260D14, C4⋊D28.5C2, C28.130(C2×D4), C14.D8⋊15C2, C2.24(D7×SD16), C14.55(C4○D8), C28.29(C4○D4), C14.Q16⋊16C2, C4.3(Q8⋊2D7), (C2×Dic7).99D4, C14.40(C2×SD16), (C22×D7).53D4, C22.216(D4×D7), C14.43(C4⋊D4), C2.16(C4⋊D28), (C2×C56).161C22, (C2×C28).280C23, (C2×D28).74C22, C2.22(SD16⋊3D7), (C2×Dic14).83C22, (D7×C2×C8)⋊7C2, (C7×C4.Q8)⋊9C2, (C2×C56⋊C2)⋊28C2, (C2×C14).285(C2×D4), (C7×C4⋊C4).73C22, (C2×C7⋊C8).228C22, (C2×C4×D7).233C22, (C2×C4).383(C22×D7), SmallGroup(448,398)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊8D28
G = < a,b,c | a8=b28=c2=1, bab-1=cac=a3, cbc=b-1 >
Subgroups: 780 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C8⋊8D4, C8×D7, C56⋊C2, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C2×D28, C14.D8, C14.Q16, C7×C4.Q8, C4⋊D28, D14⋊2Q8, D7×C2×C8, C2×C56⋊C2, C8⋊8D28
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C4○D8, D28, C22×D7, C8⋊8D4, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, D7×SD16, SD16⋊3D7, C8⋊8D28
►On 224 points
Generators in S224
(1 123 79 211 176 86 159 46)(2 212 160 124 177 47 80 87)(3 125 81 213 178 88 161 48)(4 214 162 126 179 49 82 89)(5 127 83 215 180 90 163 50)(6 216 164 128 181 51 84 91)(7 129 57 217 182 92 165 52)(8 218 166 130 183 53 58 93)(9 131 59 219 184 94 167 54)(10 220 168 132 185 55 60 95)(11 133 61 221 186 96 141 56)(12 222 142 134 187 29 62 97)(13 135 63 223 188 98 143 30)(14 224 144 136 189 31 64 99)(15 137 65 197 190 100 145 32)(16 198 146 138 191 33 66 101)(17 139 67 199 192 102 147 34)(18 200 148 140 193 35 68 103)(19 113 69 201 194 104 149 36)(20 202 150 114 195 37 70 105)(21 115 71 203 196 106 151 38)(22 204 152 116 169 39 72 107)(23 117 73 205 170 108 153 40)(24 206 154 118 171 41 74 109)(25 119 75 207 172 110 155 42)(26 208 156 120 173 43 76 111)(27 121 77 209 174 112 157 44)(28 210 158 122 175 45 78 85)
(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)
(1 175)(2 174)(3 173)(4 172)(5 171)(6 170)(7 169)(8 196)(9 195)(10 194)(11 193)(12 192)(13 191)(14 190)(15 189)(16 188)(17 187)(18 186)(19 185)(20 184)(21 183)(22 182)(23 181)(24 180)(25 179)(26 178)(27 177)(28 176)(29 199)(30 198)(31 197)(32 224)(33 223)(34 222)(35 221)(36 220)(37 219)(38 218)(39 217)(40 216)(41 215)(42 214)(43 213)(44 212)(45 211)(46 210)(47 209)(48 208)(49 207)(50 206)(51 205)(52 204)(53 203)(54 202)(55 201)(56 200)(57 152)(58 151)(59 150)(60 149)(61 148)(62 147)(63 146)(64 145)(65 144)(66 143)(67 142)(68 141)(69 168)(70 167)(71 166)(72 165)(73 164)(74 163)(75 162)(76 161)(77 160)(78 159)(79 158)(80 157)(81 156)(82 155)(83 154)(84 153)(85 123)(86 122)(87 121)(88 120)(89 119)(90 118)(91 117)(92 116)(93 115)(94 114)(95 113)(96 140)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)(105 131)(106 130)(107 129)(108 128)(109 127)(110 126)(111 125)(112 124)
G:=sub<Sym(224)| (1,123,79,211,176,86,159,46)(2,212,160,124,177,47,80,87)(3,125,81,213,178,88,161,48)(4,214,162,126,179,49,82,89)(5,127,83,215,180,90,163,50)(6,216,164,128,181,51,84,91)(7,129,57,217,182,92,165,52)(8,218,166,130,183,53,58,93)(9,131,59,219,184,94,167,54)(10,220,168,132,185,55,60,95)(11,133,61,221,186,96,141,56)(12,222,142,134,187,29,62,97)(13,135,63,223,188,98,143,30)(14,224,144,136,189,31,64,99)(15,137,65,197,190,100,145,32)(16,198,146,138,191,33,66,101)(17,139,67,199,192,102,147,34)(18,200,148,140,193,35,68,103)(19,113,69,201,194,104,149,36)(20,202,150,114,195,37,70,105)(21,115,71,203,196,106,151,38)(22,204,152,116,169,39,72,107)(23,117,73,205,170,108,153,40)(24,206,154,118,171,41,74,109)(25,119,75,207,172,110,155,42)(26,208,156,120,173,43,76,111)(27,121,77,209,174,112,157,44)(28,210,158,122,175,45,78,85), (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), (1,175)(2,174)(3,173)(4,172)(5,171)(6,170)(7,169)(8,196)(9,195)(10,194)(11,193)(12,192)(13,191)(14,190)(15,189)(16,188)(17,187)(18,186)(19,185)(20,184)(21,183)(22,182)(23,181)(24,180)(25,179)(26,178)(27,177)(28,176)(29,199)(30,198)(31,197)(32,224)(33,223)(34,222)(35,221)(36,220)(37,219)(38,218)(39,217)(40,216)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,200)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)>;
G:=Group( (1,123,79,211,176,86,159,46)(2,212,160,124,177,47,80,87)(3,125,81,213,178,88,161,48)(4,214,162,126,179,49,82,89)(5,127,83,215,180,90,163,50)(6,216,164,128,181,51,84,91)(7,129,57,217,182,92,165,52)(8,218,166,130,183,53,58,93)(9,131,59,219,184,94,167,54)(10,220,168,132,185,55,60,95)(11,133,61,221,186,96,141,56)(12,222,142,134,187,29,62,97)(13,135,63,223,188,98,143,30)(14,224,144,136,189,31,64,99)(15,137,65,197,190,100,145,32)(16,198,146,138,191,33,66,101)(17,139,67,199,192,102,147,34)(18,200,148,140,193,35,68,103)(19,113,69,201,194,104,149,36)(20,202,150,114,195,37,70,105)(21,115,71,203,196,106,151,38)(22,204,152,116,169,39,72,107)(23,117,73,205,170,108,153,40)(24,206,154,118,171,41,74,109)(25,119,75,207,172,110,155,42)(26,208,156,120,173,43,76,111)(27,121,77,209,174,112,157,44)(28,210,158,122,175,45,78,85), (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), (1,175)(2,174)(3,173)(4,172)(5,171)(6,170)(7,169)(8,196)(9,195)(10,194)(11,193)(12,192)(13,191)(14,190)(15,189)(16,188)(17,187)(18,186)(19,185)(20,184)(21,183)(22,182)(23,181)(24,180)(25,179)(26,178)(27,177)(28,176)(29,199)(30,198)(31,197)(32,224)(33,223)(34,222)(35,221)(36,220)(37,219)(38,218)(39,217)(40,216)(41,215)(42,214)(43,213)(44,212)(45,211)(46,210)(47,209)(48,208)(49,207)(50,206)(51,205)(52,204)(53,203)(54,202)(55,201)(56,200)(57,152)(58,151)(59,150)(60,149)(61,148)(62,147)(63,146)(64,145)(65,144)(66,143)(67,142)(68,141)(69,168)(70,167)(71,166)(72,165)(73,164)(74,163)(75,162)(76,161)(77,160)(78,159)(79,158)(80,157)(81,156)(82,155)(83,154)(84,153)(85,123)(86,122)(87,121)(88,120)(89,119)(90,118)(91,117)(92,116)(93,115)(94,114)(95,113)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124) );
G=PermutationGroup([[(1,123,79,211,176,86,159,46),(2,212,160,124,177,47,80,87),(3,125,81,213,178,88,161,48),(4,214,162,126,179,49,82,89),(5,127,83,215,180,90,163,50),(6,216,164,128,181,51,84,91),(7,129,57,217,182,92,165,52),(8,218,166,130,183,53,58,93),(9,131,59,219,184,94,167,54),(10,220,168,132,185,55,60,95),(11,133,61,221,186,96,141,56),(12,222,142,134,187,29,62,97),(13,135,63,223,188,98,143,30),(14,224,144,136,189,31,64,99),(15,137,65,197,190,100,145,32),(16,198,146,138,191,33,66,101),(17,139,67,199,192,102,147,34),(18,200,148,140,193,35,68,103),(19,113,69,201,194,104,149,36),(20,202,150,114,195,37,70,105),(21,115,71,203,196,106,151,38),(22,204,152,116,169,39,72,107),(23,117,73,205,170,108,153,40),(24,206,154,118,171,41,74,109),(25,119,75,207,172,110,155,42),(26,208,156,120,173,43,76,111),(27,121,77,209,174,112,157,44),(28,210,158,122,175,45,78,85)], [(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)], [(1,175),(2,174),(3,173),(4,172),(5,171),(6,170),(7,169),(8,196),(9,195),(10,194),(11,193),(12,192),(13,191),(14,190),(15,189),(16,188),(17,187),(18,186),(19,185),(20,184),(21,183),(22,182),(23,181),(24,180),(25,179),(26,178),(27,177),(28,176),(29,199),(30,198),(31,197),(32,224),(33,223),(34,222),(35,221),(36,220),(37,219),(38,218),(39,217),(40,216),(41,215),(42,214),(43,213),(44,212),(45,211),(46,210),(47,209),(48,208),(49,207),(50,206),(51,205),(52,204),(53,203),(54,202),(55,201),(56,200),(57,152),(58,151),(59,150),(60,149),(61,148),(62,147),(63,146),(64,145),(65,144),(66,143),(67,142),(68,141),(69,168),(70,167),(71,166),(72,165),(73,164),(74,163),(75,162),(76,161),(77,160),(78,159),(79,158),(80,157),(81,156),(82,155),(83,154),(84,153),(85,123),(86,122),(87,121),(88,120),(89,119),(90,118),(91,117),(92,116),(93,115),(94,114),(95,113),(96,140),(97,139),(98,138),(99,137),(100,136),(101,135),(102,134),(103,133),(104,132),(105,131),(106,130),(107,129),(108,128),(109,127),(110,126),(111,125),(112,124)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 4 | 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 | 2 | 2 | 8 | 8 | 14 | 14 | 56 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | C4○D8 | D28 | Q8⋊2D7 | D4×D7 | D7×SD16 | SD16⋊3D7 |
| kernel | C8⋊8D28 | C14.D8 | C14.Q16 | C7×C4.Q8 | C4⋊D28 | D14⋊2Q8 | D7×C2×C8 | C2×C56⋊C2 | C56 | C2×Dic7 | C22×D7 | C4.Q8 | C28 | D14 | C4⋊C4 | C2×C8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C8⋊8D28 ►in GL4(𝔽113) generated by
| 69 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 98 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 58 | 32 |
| 0 | 0 | 81 | 109 |
| 0 | 15 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 55 | 75 |
| 0 | 0 | 32 | 58 |
G:=sub<GL(4,GF(113))| [69,0,0,0,0,18,0,0,0,0,112,0,0,0,0,112],[0,98,0,0,98,0,0,0,0,0,58,81,0,0,32,109],[0,98,0,0,15,0,0,0,0,0,55,32,0,0,75,58] >;
C8⋊8D28 in GAP, Magma, Sage, TeX
C_8\rtimes_8D_{28} % in TeX
G:=Group("C8:8D28"); // GroupNames label
G:=SmallGroup(448,398);
// by ID
G=gap.SmallGroup(448,398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,555,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations