metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊1SD16, D28.18D4, C42.32D14, C4⋊C8⋊8D7, C4⋊3(C56⋊C2), C4.129(D4×D7), C28⋊2Q8⋊11C2, (C4×D28).10C2, (C2×C4).131D28, (C2×C8).128D14, C28.338(C2×D4), (C2×C28).120D4, C7⋊2(D4.D4), (C4×C28).67C22, C14.10(C2×SD16), C28.327(C4○D4), C28.44D4⋊12C2, C2.10(C4⋊D28), C14.37(C4⋊D4), (C2×C56).138C22, (C2×C28).751C23, C4.43(Q8⋊2D7), C22.114(C2×D28), C2.17(C8.D14), (C2×D28).194C22, C14.14(C8.C22), C4⋊Dic7.272C22, (C2×Dic14).14C22, (C7×C4⋊C8)⋊10C2, (C2×C56⋊C2).5C2, C2.13(C2×C56⋊C2), (C2×C14).134(C2×D4), (C2×C4).696(C22×D7), SmallGroup(448,375)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊SD16
G = < a,b,c | a28=b8=c2=1, bab-1=a15, cac=a13, cbc=b3 >
Subgroups: 740 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, D14, C2×C14, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C56, Dic14, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, D4.D4, C56⋊C2, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C28.44D4, C7×C4⋊C8, C28⋊2Q8, C4×D28, C2×C56⋊C2, C28⋊SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C8.C22, D28, C22×D7, D4.D4, C56⋊C2, C2×D28, D4×D7, Q8⋊2D7, C4⋊D28, C2×C56⋊C2, C8.D14, C28⋊SD16
►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 82 215 191 162 55 85 117)(2 69 216 178 163 42 86 132)(3 84 217 193 164 29 87 119)(4 71 218 180 165 44 88 134)(5 58 219 195 166 31 89 121)(6 73 220 182 167 46 90 136)(7 60 221 169 168 33 91 123)(8 75 222 184 141 48 92 138)(9 62 223 171 142 35 93 125)(10 77 224 186 143 50 94 140)(11 64 197 173 144 37 95 127)(12 79 198 188 145 52 96 114)(13 66 199 175 146 39 97 129)(14 81 200 190 147 54 98 116)(15 68 201 177 148 41 99 131)(16 83 202 192 149 56 100 118)(17 70 203 179 150 43 101 133)(18 57 204 194 151 30 102 120)(19 72 205 181 152 45 103 135)(20 59 206 196 153 32 104 122)(21 74 207 183 154 47 105 137)(22 61 208 170 155 34 106 124)(23 76 209 185 156 49 107 139)(24 63 210 172 157 36 108 126)(25 78 211 187 158 51 109 113)(26 65 212 174 159 38 110 128)(27 80 213 189 160 53 111 115)(28 67 214 176 161 40 112 130)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 129)(30 114)(31 127)(32 140)(33 125)(34 138)(35 123)(36 136)(37 121)(38 134)(39 119)(40 132)(41 117)(42 130)(43 115)(44 128)(45 113)(46 126)(47 139)(48 124)(49 137)(50 122)(51 135)(52 120)(53 133)(54 118)(55 131)(56 116)(57 188)(58 173)(59 186)(60 171)(61 184)(62 169)(63 182)(64 195)(65 180)(66 193)(67 178)(68 191)(69 176)(70 189)(71 174)(72 187)(73 172)(74 185)(75 170)(76 183)(77 196)(78 181)(79 194)(80 179)(81 192)(82 177)(83 190)(84 175)(85 201)(86 214)(87 199)(88 212)(89 197)(90 210)(91 223)(92 208)(93 221)(94 206)(95 219)(96 204)(97 217)(98 202)(99 215)(100 200)(101 213)(102 198)(103 211)(104 224)(105 209)(106 222)(107 207)(108 220)(109 205)(110 218)(111 203)(112 216)(141 155)(142 168)(143 153)(144 166)(145 151)(146 164)(147 149)(148 162)(150 160)(152 158)(154 156)(157 167)(159 165)(161 163)
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), (1,82,215,191,162,55,85,117)(2,69,216,178,163,42,86,132)(3,84,217,193,164,29,87,119)(4,71,218,180,165,44,88,134)(5,58,219,195,166,31,89,121)(6,73,220,182,167,46,90,136)(7,60,221,169,168,33,91,123)(8,75,222,184,141,48,92,138)(9,62,223,171,142,35,93,125)(10,77,224,186,143,50,94,140)(11,64,197,173,144,37,95,127)(12,79,198,188,145,52,96,114)(13,66,199,175,146,39,97,129)(14,81,200,190,147,54,98,116)(15,68,201,177,148,41,99,131)(16,83,202,192,149,56,100,118)(17,70,203,179,150,43,101,133)(18,57,204,194,151,30,102,120)(19,72,205,181,152,45,103,135)(20,59,206,196,153,32,104,122)(21,74,207,183,154,47,105,137)(22,61,208,170,155,34,106,124)(23,76,209,185,156,49,107,139)(24,63,210,172,157,36,108,126)(25,78,211,187,158,51,109,113)(26,65,212,174,159,38,110,128)(27,80,213,189,160,53,111,115)(28,67,214,176,161,40,112,130), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,129)(30,114)(31,127)(32,140)(33,125)(34,138)(35,123)(36,136)(37,121)(38,134)(39,119)(40,132)(41,117)(42,130)(43,115)(44,128)(45,113)(46,126)(47,139)(48,124)(49,137)(50,122)(51,135)(52,120)(53,133)(54,118)(55,131)(56,116)(57,188)(58,173)(59,186)(60,171)(61,184)(62,169)(63,182)(64,195)(65,180)(66,193)(67,178)(68,191)(69,176)(70,189)(71,174)(72,187)(73,172)(74,185)(75,170)(76,183)(77,196)(78,181)(79,194)(80,179)(81,192)(82,177)(83,190)(84,175)(85,201)(86,214)(87,199)(88,212)(89,197)(90,210)(91,223)(92,208)(93,221)(94,206)(95,219)(96,204)(97,217)(98,202)(99,215)(100,200)(101,213)(102,198)(103,211)(104,224)(105,209)(106,222)(107,207)(108,220)(109,205)(110,218)(111,203)(112,216)(141,155)(142,168)(143,153)(144,166)(145,151)(146,164)(147,149)(148,162)(150,160)(152,158)(154,156)(157,167)(159,165)(161,163)>;
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), (1,82,215,191,162,55,85,117)(2,69,216,178,163,42,86,132)(3,84,217,193,164,29,87,119)(4,71,218,180,165,44,88,134)(5,58,219,195,166,31,89,121)(6,73,220,182,167,46,90,136)(7,60,221,169,168,33,91,123)(8,75,222,184,141,48,92,138)(9,62,223,171,142,35,93,125)(10,77,224,186,143,50,94,140)(11,64,197,173,144,37,95,127)(12,79,198,188,145,52,96,114)(13,66,199,175,146,39,97,129)(14,81,200,190,147,54,98,116)(15,68,201,177,148,41,99,131)(16,83,202,192,149,56,100,118)(17,70,203,179,150,43,101,133)(18,57,204,194,151,30,102,120)(19,72,205,181,152,45,103,135)(20,59,206,196,153,32,104,122)(21,74,207,183,154,47,105,137)(22,61,208,170,155,34,106,124)(23,76,209,185,156,49,107,139)(24,63,210,172,157,36,108,126)(25,78,211,187,158,51,109,113)(26,65,212,174,159,38,110,128)(27,80,213,189,160,53,111,115)(28,67,214,176,161,40,112,130), (1,15)(2,28)(3,13)(4,26)(5,11)(6,24)(7,9)(8,22)(10,20)(12,18)(14,16)(17,27)(19,25)(21,23)(29,129)(30,114)(31,127)(32,140)(33,125)(34,138)(35,123)(36,136)(37,121)(38,134)(39,119)(40,132)(41,117)(42,130)(43,115)(44,128)(45,113)(46,126)(47,139)(48,124)(49,137)(50,122)(51,135)(52,120)(53,133)(54,118)(55,131)(56,116)(57,188)(58,173)(59,186)(60,171)(61,184)(62,169)(63,182)(64,195)(65,180)(66,193)(67,178)(68,191)(69,176)(70,189)(71,174)(72,187)(73,172)(74,185)(75,170)(76,183)(77,196)(78,181)(79,194)(80,179)(81,192)(82,177)(83,190)(84,175)(85,201)(86,214)(87,199)(88,212)(89,197)(90,210)(91,223)(92,208)(93,221)(94,206)(95,219)(96,204)(97,217)(98,202)(99,215)(100,200)(101,213)(102,198)(103,211)(104,224)(105,209)(106,222)(107,207)(108,220)(109,205)(110,218)(111,203)(112,216)(141,155)(142,168)(143,153)(144,166)(145,151)(146,164)(147,149)(148,162)(150,160)(152,158)(154,156)(157,167)(159,165)(161,163) );
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)], [(1,82,215,191,162,55,85,117),(2,69,216,178,163,42,86,132),(3,84,217,193,164,29,87,119),(4,71,218,180,165,44,88,134),(5,58,219,195,166,31,89,121),(6,73,220,182,167,46,90,136),(7,60,221,169,168,33,91,123),(8,75,222,184,141,48,92,138),(9,62,223,171,142,35,93,125),(10,77,224,186,143,50,94,140),(11,64,197,173,144,37,95,127),(12,79,198,188,145,52,96,114),(13,66,199,175,146,39,97,129),(14,81,200,190,147,54,98,116),(15,68,201,177,148,41,99,131),(16,83,202,192,149,56,100,118),(17,70,203,179,150,43,101,133),(18,57,204,194,151,30,102,120),(19,72,205,181,152,45,103,135),(20,59,206,196,153,32,104,122),(21,74,207,183,154,47,105,137),(22,61,208,170,155,34,106,124),(23,76,209,185,156,49,107,139),(24,63,210,172,157,36,108,126),(25,78,211,187,158,51,109,113),(26,65,212,174,159,38,110,128),(27,80,213,189,160,53,111,115),(28,67,214,176,161,40,112,130)], [(1,15),(2,28),(3,13),(4,26),(5,11),(6,24),(7,9),(8,22),(10,20),(12,18),(14,16),(17,27),(19,25),(21,23),(29,129),(30,114),(31,127),(32,140),(33,125),(34,138),(35,123),(36,136),(37,121),(38,134),(39,119),(40,132),(41,117),(42,130),(43,115),(44,128),(45,113),(46,126),(47,139),(48,124),(49,137),(50,122),(51,135),(52,120),(53,133),(54,118),(55,131),(56,116),(57,188),(58,173),(59,186),(60,171),(61,184),(62,169),(63,182),(64,195),(65,180),(66,193),(67,178),(68,191),(69,176),(70,189),(71,174),(72,187),(73,172),(74,185),(75,170),(76,183),(77,196),(78,181),(79,194),(80,179),(81,192),(82,177),(83,190),(84,175),(85,201),(86,214),(87,199),(88,212),(89,197),(90,210),(91,223),(92,208),(93,221),(94,206),(95,219),(96,204),(97,217),(98,202),(99,215),(100,200),(101,213),(102,198),(103,211),(104,224),(105,209),(106,222),(107,207),(108,220),(109,205),(110,218),(111,203),(112,216),(141,155),(142,168),(143,153),(144,166),(145,151),(146,164),(147,149),(148,162),(150,160),(152,158),(154,156),(157,167),(159,165),(161,163)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | D28 | C56⋊C2 | C8.C22 | D4×D7 | Q8⋊2D7 | C8.D14 |
| kernel | C28⋊SD16 | C28.44D4 | C7×C4⋊C8 | C28⋊2Q8 | C4×D28 | C2×C56⋊C2 | D28 | C2×C28 | C4⋊C8 | C28 | C28 | C42 | C2×C8 | C2×C4 | C4 | C14 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 24 | 1 | 3 | 3 | 6 |
Matrix representation of C28⋊SD16 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 89 | 0 | 0 |
| 0 | 0 | 24 | 103 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 64 |
| 0 | 0 | 0 | 0 | 19 | 79 |
| 100 | 13 | 0 | 0 | 0 | 0 |
| 100 | 100 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 8 | 0 | 0 |
| 0 | 0 | 105 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 69 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,89,103,0,0,0,0,0,0,34,19,0,0,0,0,64,79],[100,100,0,0,0,0,13,100,0,0,0,0,0,0,17,105,0,0,0,0,8,96,0,0,0,0,0,0,112,0,0,0,0,0,69,1],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;
C28⋊SD16 in GAP, Magma, Sage, TeX
C_{28}\rtimes {\rm SD}_{16} % in TeX
G:=Group("C28:SD16"); // GroupNames label
G:=SmallGroup(448,375);
// by ID
G=gap.SmallGroup(448,375);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^15,c*a*c=a^13,c*b*c=b^3>;
// generators/relations