metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊3C8, C42.201D14, C7⋊3(C8×D4), C7⋊C8⋊30D4, C4⋊1(C8×D7), C28⋊2(C2×C8), C4⋊C8⋊18D7, D14⋊3(C2×C8), D14⋊C8⋊22C2, (C4×D28).7C2, C4.205(D4×D7), C14.46(C4×D4), D14⋊C4.11C4, (C2×D28).12C4, C28.364(C2×D4), (C2×C8).216D14, C4⋊Dic7.18C4, C14.28(C8○D4), C2.4(D28.C4), C14.11(C22×C8), (C4×C28).60C22, C28.334(C4○D4), C2.2(D28⋊C4), (C2×C28).831C23, (C2×C56).210C22, C4.53(Q8⋊2D7), (C4×C7⋊C8)⋊4C2, (D7×C2×C8)⋊22C2, (C7×C4⋊C8)⋊15C2, C2.13(D7×C2×C8), (C2×C4).72(C4×D7), C22.48(C2×C4×D7), (C2×C28).158(C2×C4), (C2×C7⋊C8).306C22, (C2×C4×D7).277C22, (C2×C14).86(C22×C4), (C2×Dic7).53(C2×C4), (C22×D7).37(C2×C4), (C2×C4).773(C22×D7), SmallGroup(448,368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊C8
G = < a,b,c | a28=b2=c8=1, bab=a-1, cac-1=a15, cbc-1=a14b >
Subgroups: 580 in 134 conjugacy classes, 61 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, C28, D14, D14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C7⋊C8, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×C28, C22×D7, C8×D4, C8×D7, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C4×C7⋊C8, D14⋊C8, C7×C4⋊C8, C4×D28, D7×C2×C8, D28⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C2×C8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C22×C8, C8○D4, C4×D7, C22×D7, C8×D4, C8×D7, C2×C4×D7, D4×D7, Q8⋊2D7, D28⋊C4, D7×C2×C8, D28.C4, D28⋊C8
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 180)(121 179)(122 178)(123 177)(124 176)(125 175)(126 174)(127 173)(128 172)(129 171)(130 170)(131 169)(132 196)(133 195)(134 194)(135 193)(136 192)(137 191)(138 190)(139 189)(140 188)(141 216)(142 215)(143 214)(144 213)(145 212)(146 211)(147 210)(148 209)(149 208)(150 207)(151 206)(152 205)(153 204)(154 203)(155 202)(156 201)(157 200)(158 199)(159 198)(160 197)(161 224)(162 223)(163 222)(164 221)(165 220)(166 219)(167 218)(168 217)
(1 207 68 195 42 158 104 113)(2 222 69 182 43 145 105 128)(3 209 70 169 44 160 106 115)(4 224 71 184 45 147 107 130)(5 211 72 171 46 162 108 117)(6 198 73 186 47 149 109 132)(7 213 74 173 48 164 110 119)(8 200 75 188 49 151 111 134)(9 215 76 175 50 166 112 121)(10 202 77 190 51 153 85 136)(11 217 78 177 52 168 86 123)(12 204 79 192 53 155 87 138)(13 219 80 179 54 142 88 125)(14 206 81 194 55 157 89 140)(15 221 82 181 56 144 90 127)(16 208 83 196 29 159 91 114)(17 223 84 183 30 146 92 129)(18 210 57 170 31 161 93 116)(19 197 58 185 32 148 94 131)(20 212 59 172 33 163 95 118)(21 199 60 187 34 150 96 133)(22 214 61 174 35 165 97 120)(23 201 62 189 36 152 98 135)(24 216 63 176 37 167 99 122)(25 203 64 191 38 154 100 137)(26 218 65 178 39 141 101 124)(27 205 66 193 40 156 102 139)(28 220 67 180 41 143 103 126)
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,180)(121,179)(122,178)(123,177)(124,176)(125,175)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,210)(148,209)(149,208)(150,207)(151,206)(152,205)(153,204)(154,203)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,224)(162,223)(163,222)(164,221)(165,220)(166,219)(167,218)(168,217), (1,207,68,195,42,158,104,113)(2,222,69,182,43,145,105,128)(3,209,70,169,44,160,106,115)(4,224,71,184,45,147,107,130)(5,211,72,171,46,162,108,117)(6,198,73,186,47,149,109,132)(7,213,74,173,48,164,110,119)(8,200,75,188,49,151,111,134)(9,215,76,175,50,166,112,121)(10,202,77,190,51,153,85,136)(11,217,78,177,52,168,86,123)(12,204,79,192,53,155,87,138)(13,219,80,179,54,142,88,125)(14,206,81,194,55,157,89,140)(15,221,82,181,56,144,90,127)(16,208,83,196,29,159,91,114)(17,223,84,183,30,146,92,129)(18,210,57,170,31,161,93,116)(19,197,58,185,32,148,94,131)(20,212,59,172,33,163,95,118)(21,199,60,187,34,150,96,133)(22,214,61,174,35,165,97,120)(23,201,62,189,36,152,98,135)(24,216,63,176,37,167,99,122)(25,203,64,191,38,154,100,137)(26,218,65,178,39,141,101,124)(27,205,66,193,40,156,102,139)(28,220,67,180,41,143,103,126)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(57,93)(58,92)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,180)(121,179)(122,178)(123,177)(124,176)(125,175)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)(132,196)(133,195)(134,194)(135,193)(136,192)(137,191)(138,190)(139,189)(140,188)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,210)(148,209)(149,208)(150,207)(151,206)(152,205)(153,204)(154,203)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,224)(162,223)(163,222)(164,221)(165,220)(166,219)(167,218)(168,217), (1,207,68,195,42,158,104,113)(2,222,69,182,43,145,105,128)(3,209,70,169,44,160,106,115)(4,224,71,184,45,147,107,130)(5,211,72,171,46,162,108,117)(6,198,73,186,47,149,109,132)(7,213,74,173,48,164,110,119)(8,200,75,188,49,151,111,134)(9,215,76,175,50,166,112,121)(10,202,77,190,51,153,85,136)(11,217,78,177,52,168,86,123)(12,204,79,192,53,155,87,138)(13,219,80,179,54,142,88,125)(14,206,81,194,55,157,89,140)(15,221,82,181,56,144,90,127)(16,208,83,196,29,159,91,114)(17,223,84,183,30,146,92,129)(18,210,57,170,31,161,93,116)(19,197,58,185,32,148,94,131)(20,212,59,172,33,163,95,118)(21,199,60,187,34,150,96,133)(22,214,61,174,35,165,97,120)(23,201,62,189,36,152,98,135)(24,216,63,176,37,167,99,122)(25,203,64,191,38,154,100,137)(26,218,65,178,39,141,101,124)(27,205,66,193,40,156,102,139)(28,220,67,180,41,143,103,126) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(57,93),(58,92),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(113,187),(114,186),(115,185),(116,184),(117,183),(118,182),(119,181),(120,180),(121,179),(122,178),(123,177),(124,176),(125,175),(126,174),(127,173),(128,172),(129,171),(130,170),(131,169),(132,196),(133,195),(134,194),(135,193),(136,192),(137,191),(138,190),(139,189),(140,188),(141,216),(142,215),(143,214),(144,213),(145,212),(146,211),(147,210),(148,209),(149,208),(150,207),(151,206),(152,205),(153,204),(154,203),(155,202),(156,201),(157,200),(158,199),(159,198),(160,197),(161,224),(162,223),(163,222),(164,221),(165,220),(166,219),(167,218),(168,217)], [(1,207,68,195,42,158,104,113),(2,222,69,182,43,145,105,128),(3,209,70,169,44,160,106,115),(4,224,71,184,45,147,107,130),(5,211,72,171,46,162,108,117),(6,198,73,186,47,149,109,132),(7,213,74,173,48,164,110,119),(8,200,75,188,49,151,111,134),(9,215,76,175,50,166,112,121),(10,202,77,190,51,153,85,136),(11,217,78,177,52,168,86,123),(12,204,79,192,53,155,87,138),(13,219,80,179,54,142,88,125),(14,206,81,194,55,157,89,140),(15,221,82,181,56,144,90,127),(16,208,83,196,29,159,91,114),(17,223,84,183,30,146,92,129),(18,210,57,170,31,161,93,116),(19,197,58,185,32,148,94,131),(20,212,59,172,33,163,95,118),(21,199,60,187,34,150,96,133),(22,214,61,174,35,165,97,120),(23,201,62,189,36,152,98,135),(24,216,63,176,37,167,99,122),(25,203,64,191,38,154,100,137),(26,218,65,178,39,141,101,124),(27,205,66,193,40,156,102,139),(28,220,67,180,41,143,103,126)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | D7 | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C8×D7 | D4×D7 | Q8⋊2D7 | D28.C4 |
| kernel | D28⋊C8 | C4×C7⋊C8 | D14⋊C8 | C7×C4⋊C8 | C4×D28 | D7×C2×C8 | C4⋊Dic7 | D14⋊C4 | C2×D28 | D28 | C7⋊C8 | C4⋊C8 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 16 | 2 | 3 | 2 | 3 | 6 | 4 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of D28⋊C8 ►in GL4(𝔽113) generated by
| 25 | 59 | 0 | 0 |
| 89 | 79 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 0 | 79 | 0 | 0 |
| 103 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 1 |
| 44 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [25,89,0,0,59,79,0,0,0,0,0,112,0,0,1,0],[0,103,0,0,79,0,0,0,0,0,112,0,0,0,0,1],[44,0,0,0,0,44,0,0,0,0,0,1,0,0,1,0] >;
D28⋊C8 in GAP, Magma, Sage, TeX
D_{28}\rtimes C_8 % in TeX
G:=Group("D28:C8"); // GroupNames label
G:=SmallGroup(448,368);
// by ID
G=gap.SmallGroup(448,368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^15,c*b*c^-1=a^14*b>;
// generators/relations