metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊9D28, C56⋊16D4, D14⋊1M4(2), C42.15D14, C8⋊C4⋊8D7, C7⋊1(C8⋊9D4), C28⋊C8⋊3C2, D14⋊C8⋊36C2, D14⋊C4.4C4, (C4×D28).4C2, (C2×D28).7C4, C14.10(C4×D4), C2.13(C4×D28), C4.77(C2×D28), C4⋊Dic7.9C4, C28.297(C2×D4), (C2×C8).157D14, C14.21(C8○D4), C2.7(D28.C4), (C4×C28).13C22, C2.10(D7×M4(2)), C4.130(C4○D28), C28.246(C4○D4), (C2×C28).812C23, (C2×C56).226C22, C14.17(C2×M4(2)), (D7×C2×C8)⋊26C2, (C7×C8⋊C4)⋊7C2, (C2×C4).29(C4×D7), (C2×C8⋊D7)⋊24C2, C22.99(C2×C4×D7), (C2×C28).37(C2×C4), (C2×C7⋊C8).295C22, (C2×C4×D7).177C22, (C2×C14).67(C22×C4), (C2×Dic7).14(C2×C4), (C22×D7).33(C2×C4), (C2×C4).754(C22×D7), SmallGroup(448,240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊9D28
G = < a,b,c | a8=b28=c2=1, bab-1=cac=a5, cbc=b-1 >
Subgroups: 580 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, 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, D14, D14, C2×C14, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C7⋊C8, C56, C56, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C8⋊9D4, C8×D7, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C28⋊C8, D14⋊C8, C7×C8⋊C4, C4×D28, D7×C2×C8, C2×C8⋊D7, C8⋊9D28
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⋊9D4, C2×C4×D7, C2×D28, C4○D28, C4×D28, D7×M4(2), D28.C4, C8⋊9D28
►On 224 points
Generators in S224
(1 196 163 30 71 114 94 217)(2 115 164 218 72 169 95 31)(3 170 165 32 73 116 96 219)(4 117 166 220 74 171 97 33)(5 172 167 34 75 118 98 221)(6 119 168 222 76 173 99 35)(7 174 141 36 77 120 100 223)(8 121 142 224 78 175 101 37)(9 176 143 38 79 122 102 197)(10 123 144 198 80 177 103 39)(11 178 145 40 81 124 104 199)(12 125 146 200 82 179 105 41)(13 180 147 42 83 126 106 201)(14 127 148 202 84 181 107 43)(15 182 149 44 57 128 108 203)(16 129 150 204 58 183 109 45)(17 184 151 46 59 130 110 205)(18 131 152 206 60 185 111 47)(19 186 153 48 61 132 112 207)(20 133 154 208 62 187 85 49)(21 188 155 50 63 134 86 209)(22 135 156 210 64 189 87 51)(23 190 157 52 65 136 88 211)(24 137 158 212 66 191 89 53)(25 192 159 54 67 138 90 213)(26 139 160 214 68 193 91 55)(27 194 161 56 69 140 92 215)(28 113 162 216 70 195 93 29)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 30)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(103 112)(104 111)(105 110)(106 109)(107 108)(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)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)(169 194)(170 193)(171 192)(172 191)(173 190)(174 189)(175 188)(176 187)(177 186)(178 185)(179 184)(180 183)(181 182)(195 196)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)(209 224)(210 223)(211 222)(212 221)(213 220)(214 219)(215 218)(216 217)
G:=sub<Sym(224)| (1,196,163,30,71,114,94,217)(2,115,164,218,72,169,95,31)(3,170,165,32,73,116,96,219)(4,117,166,220,74,171,97,33)(5,172,167,34,75,118,98,221)(6,119,168,222,76,173,99,35)(7,174,141,36,77,120,100,223)(8,121,142,224,78,175,101,37)(9,176,143,38,79,122,102,197)(10,123,144,198,80,177,103,39)(11,178,145,40,81,124,104,199)(12,125,146,200,82,179,105,41)(13,180,147,42,83,126,106,201)(14,127,148,202,84,181,107,43)(15,182,149,44,57,128,108,203)(16,129,150,204,58,183,109,45)(17,184,151,46,59,130,110,205)(18,131,152,206,60,185,111,47)(19,186,153,48,61,132,112,207)(20,133,154,208,62,187,85,49)(21,188,155,50,63,134,86,209)(22,135,156,210,64,189,87,51)(23,190,157,52,65,136,88,211)(24,137,158,212,66,191,89,53)(25,192,159,54,67,138,90,213)(26,139,160,214,68,193,91,55)(27,194,161,56,69,140,92,215)(28,113,162,216,70,195,93,29), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(103,112)(104,111)(105,110)(106,109)(107,108)(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)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(169,194)(170,193)(171,192)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)(195,196)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)>;
G:=Group( (1,196,163,30,71,114,94,217)(2,115,164,218,72,169,95,31)(3,170,165,32,73,116,96,219)(4,117,166,220,74,171,97,33)(5,172,167,34,75,118,98,221)(6,119,168,222,76,173,99,35)(7,174,141,36,77,120,100,223)(8,121,142,224,78,175,101,37)(9,176,143,38,79,122,102,197)(10,123,144,198,80,177,103,39)(11,178,145,40,81,124,104,199)(12,125,146,200,82,179,105,41)(13,180,147,42,83,126,106,201)(14,127,148,202,84,181,107,43)(15,182,149,44,57,128,108,203)(16,129,150,204,58,183,109,45)(17,184,151,46,59,130,110,205)(18,131,152,206,60,185,111,47)(19,186,153,48,61,132,112,207)(20,133,154,208,62,187,85,49)(21,188,155,50,63,134,86,209)(22,135,156,210,64,189,87,51)(23,190,157,52,65,136,88,211)(24,137,158,212,66,191,89,53)(25,192,159,54,67,138,90,213)(26,139,160,214,68,193,91,55)(27,194,161,56,69,140,92,215)(28,113,162,216,70,195,93,29), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(103,112)(104,111)(105,110)(106,109)(107,108)(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)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(169,194)(170,193)(171,192)(172,191)(173,190)(174,189)(175,188)(176,187)(177,186)(178,185)(179,184)(180,183)(181,182)(195,196)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,224)(210,223)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217) );
G=PermutationGroup([[(1,196,163,30,71,114,94,217),(2,115,164,218,72,169,95,31),(3,170,165,32,73,116,96,219),(4,117,166,220,74,171,97,33),(5,172,167,34,75,118,98,221),(6,119,168,222,76,173,99,35),(7,174,141,36,77,120,100,223),(8,121,142,224,78,175,101,37),(9,176,143,38,79,122,102,197),(10,123,144,198,80,177,103,39),(11,178,145,40,81,124,104,199),(12,125,146,200,82,179,105,41),(13,180,147,42,83,126,106,201),(14,127,148,202,84,181,107,43),(15,182,149,44,57,128,108,203),(16,129,150,204,58,183,109,45),(17,184,151,46,59,130,110,205),(18,131,152,206,60,185,111,47),(19,186,153,48,61,132,112,207),(20,133,154,208,62,187,85,49),(21,188,155,50,63,134,86,209),(22,135,156,210,64,189,87,51),(23,190,157,52,65,136,88,211),(24,137,158,212,66,191,89,53),(25,192,159,54,67,138,90,213),(26,139,160,214,68,193,91,55),(27,194,161,56,69,140,92,215),(28,113,162,216,70,195,93,29)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,30),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(85,102),(86,101),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(103,112),(104,111),(105,110),(106,109),(107,108),(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),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,168),(158,167),(159,166),(160,165),(161,164),(162,163),(169,194),(170,193),(171,192),(172,191),(173,190),(174,189),(175,188),(176,187),(177,186),(178,185),(179,184),(180,183),(181,182),(195,196),(197,208),(198,207),(199,206),(200,205),(201,204),(202,203),(209,224),(210,223),(211,222),(212,221),(213,220),(214,219),(215,218),(216,217)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | C4○D4 | M4(2) | D14 | D14 | C8○D4 | D28 | C4×D7 | C4○D28 | D7×M4(2) | D28.C4 |
| kernel | C8⋊9D28 | C28⋊C8 | D14⋊C8 | C7×C8⋊C4 | C4×D28 | D7×C2×C8 | C2×C8⋊D7 | C4⋊Dic7 | D14⋊C4 | C2×D28 | C56 | C8⋊C4 | C28 | D14 | C42 | C2×C8 | C14 | C8 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 3 | 2 | 4 | 3 | 6 | 4 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C8⋊9D28 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 47 | 95 |
| 94 | 13 | 0 | 0 |
| 100 | 104 | 0 | 0 |
| 0 | 0 | 6 | 106 |
| 0 | 0 | 86 | 107 |
| 94 | 13 | 0 | 0 |
| 94 | 19 | 0 | 0 |
| 0 | 0 | 6 | 106 |
| 0 | 0 | 5 | 107 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,18,47,0,0,0,95],[94,100,0,0,13,104,0,0,0,0,6,86,0,0,106,107],[94,94,0,0,13,19,0,0,0,0,6,5,0,0,106,107] >;
C8⋊9D28 in GAP, Magma, Sage, TeX
C_8\rtimes_9D_{28} % in TeX
G:=Group("C8:9D28"); // GroupNames label
G:=SmallGroup(448,240);
// by ID
G=gap.SmallGroup(448,240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=b^-1>;
// generators/relations