direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C28, (C4×C8)⋊7C14, C8⋊4(C2×C28), (C4×C56)⋊23C2, C56⋊26(C2×C4), D4⋊1(C2×C28), (C4×D4)⋊1C14, C2.3(C14×D8), (D4×C28)⋊30C2, (C2×D8).7C14, C2.D8⋊14C14, C2.12(D4×C28), C14.75(C2×D8), (C14×D8).14C2, C14.114(C4×D4), (C2×C28).360D4, C4.9(C22×C28), D4⋊C4⋊21C14, C42.70(C2×C14), C22.51(D4×C14), C14.116(C4○D8), C28.256(C4○D4), C28.154(C22×C4), (C2×C28).904C23, (C2×C56).421C22, (C4×C28).355C22, (D4×C14).290C22, C2.3(C7×C4○D8), C4.1(C7×C4○D4), (C7×D4)⋊14(C2×C4), (C7×C2.D8)⋊29C2, (C2×C4).50(C7×D4), C4⋊C4.45(C2×C14), (C2×C8).65(C2×C14), (C7×D4⋊C4)⋊44C2, (C2×D4).48(C2×C14), (C2×C14).627(C2×D4), (C7×C4⋊C4).366C22, (C2×C4).79(C22×C14), SmallGroup(448,845)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C28
G = < a,b,c | a28=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 250 in 134 conjugacy classes, 74 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C28, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, C4×D8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C22×C28, D4×C14, C4×C56, C7×D4⋊C4, C7×C2.D8, D4×C28, C14×D8, D8×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, D8, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×D8, C4○D8, C2×C28, C7×D4, C22×C14, C4×D8, C7×D8, C22×C28, D4×C14, C7×C4○D4, D4×C28, C14×D8, C7×C4○D8, D8×C28
►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)
(1 206 87 41 115 72 142 182)(2 207 88 42 116 73 143 183)(3 208 89 43 117 74 144 184)(4 209 90 44 118 75 145 185)(5 210 91 45 119 76 146 186)(6 211 92 46 120 77 147 187)(7 212 93 47 121 78 148 188)(8 213 94 48 122 79 149 189)(9 214 95 49 123 80 150 190)(10 215 96 50 124 81 151 191)(11 216 97 51 125 82 152 192)(12 217 98 52 126 83 153 193)(13 218 99 53 127 84 154 194)(14 219 100 54 128 57 155 195)(15 220 101 55 129 58 156 196)(16 221 102 56 130 59 157 169)(17 222 103 29 131 60 158 170)(18 223 104 30 132 61 159 171)(19 224 105 31 133 62 160 172)(20 197 106 32 134 63 161 173)(21 198 107 33 135 64 162 174)(22 199 108 34 136 65 163 175)(23 200 109 35 137 66 164 176)(24 201 110 36 138 67 165 177)(25 202 111 37 139 68 166 178)(26 203 112 38 140 69 167 179)(27 204 85 39 113 70 168 180)(28 205 86 40 114 71 141 181)
(1 182)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 193)(13 194)(14 195)(15 196)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 176)(24 177)(25 178)(26 179)(27 180)(28 181)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 85)(71 86)(72 87)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(141 205)(142 206)(143 207)(144 208)(145 209)(146 210)(147 211)(148 212)(149 213)(150 214)(151 215)(152 216)(153 217)(154 218)(155 219)(156 220)(157 221)(158 222)(159 223)(160 224)(161 197)(162 198)(163 199)(164 200)(165 201)(166 202)(167 203)(168 204)
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), (1,206,87,41,115,72,142,182)(2,207,88,42,116,73,143,183)(3,208,89,43,117,74,144,184)(4,209,90,44,118,75,145,185)(5,210,91,45,119,76,146,186)(6,211,92,46,120,77,147,187)(7,212,93,47,121,78,148,188)(8,213,94,48,122,79,149,189)(9,214,95,49,123,80,150,190)(10,215,96,50,124,81,151,191)(11,216,97,51,125,82,152,192)(12,217,98,52,126,83,153,193)(13,218,99,53,127,84,154,194)(14,219,100,54,128,57,155,195)(15,220,101,55,129,58,156,196)(16,221,102,56,130,59,157,169)(17,222,103,29,131,60,158,170)(18,223,104,30,132,61,159,171)(19,224,105,31,133,62,160,172)(20,197,106,32,134,63,161,173)(21,198,107,33,135,64,162,174)(22,199,108,34,136,65,163,175)(23,200,109,35,137,66,164,176)(24,201,110,36,138,67,165,177)(25,202,111,37,139,68,166,178)(26,203,112,38,140,69,167,179)(27,204,85,39,113,70,168,180)(28,205,86,40,114,71,141,181), (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)(161,197)(162,198)(163,199)(164,200)(165,201)(166,202)(167,203)(168,204)>;
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), (1,206,87,41,115,72,142,182)(2,207,88,42,116,73,143,183)(3,208,89,43,117,74,144,184)(4,209,90,44,118,75,145,185)(5,210,91,45,119,76,146,186)(6,211,92,46,120,77,147,187)(7,212,93,47,121,78,148,188)(8,213,94,48,122,79,149,189)(9,214,95,49,123,80,150,190)(10,215,96,50,124,81,151,191)(11,216,97,51,125,82,152,192)(12,217,98,52,126,83,153,193)(13,218,99,53,127,84,154,194)(14,219,100,54,128,57,155,195)(15,220,101,55,129,58,156,196)(16,221,102,56,130,59,157,169)(17,222,103,29,131,60,158,170)(18,223,104,30,132,61,159,171)(19,224,105,31,133,62,160,172)(20,197,106,32,134,63,161,173)(21,198,107,33,135,64,162,174)(22,199,108,34,136,65,163,175)(23,200,109,35,137,66,164,176)(24,201,110,36,138,67,165,177)(25,202,111,37,139,68,166,178)(26,203,112,38,140,69,167,179)(27,204,85,39,113,70,168,180)(28,205,86,40,114,71,141,181), (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,85)(71,86)(72,87)(73,88)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,211)(148,212)(149,213)(150,214)(151,215)(152,216)(153,217)(154,218)(155,219)(156,220)(157,221)(158,222)(159,223)(160,224)(161,197)(162,198)(163,199)(164,200)(165,201)(166,202)(167,203)(168,204) );
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)], [(1,206,87,41,115,72,142,182),(2,207,88,42,116,73,143,183),(3,208,89,43,117,74,144,184),(4,209,90,44,118,75,145,185),(5,210,91,45,119,76,146,186),(6,211,92,46,120,77,147,187),(7,212,93,47,121,78,148,188),(8,213,94,48,122,79,149,189),(9,214,95,49,123,80,150,190),(10,215,96,50,124,81,151,191),(11,216,97,51,125,82,152,192),(12,217,98,52,126,83,153,193),(13,218,99,53,127,84,154,194),(14,219,100,54,128,57,155,195),(15,220,101,55,129,58,156,196),(16,221,102,56,130,59,157,169),(17,222,103,29,131,60,158,170),(18,223,104,30,132,61,159,171),(19,224,105,31,133,62,160,172),(20,197,106,32,134,63,161,173),(21,198,107,33,135,64,162,174),(22,199,108,34,136,65,163,175),(23,200,109,35,137,66,164,176),(24,201,110,36,138,67,165,177),(25,202,111,37,139,68,166,178),(26,203,112,38,140,69,167,179),(27,204,85,39,113,70,168,180),(28,205,86,40,114,71,141,181)], [(1,182),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,193),(13,194),(14,195),(15,196),(16,169),(17,170),(18,171),(19,172),(20,173),(21,174),(22,175),(23,176),(24,177),(25,178),(26,179),(27,180),(28,181),(29,131),(30,132),(31,133),(32,134),(33,135),(34,136),(35,137),(36,138),(37,139),(38,140),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,100),(58,101),(59,102),(60,103),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,111),(69,112),(70,85),(71,86),(72,87),(73,88),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(141,205),(142,206),(143,207),(144,208),(145,209),(146,210),(147,211),(148,212),(149,213),(150,214),(151,215),(152,216),(153,217),(154,218),(155,219),(156,220),(157,221),(158,222),(159,223),(160,224),(161,197),(162,198),(163,199),(164,200),(165,201),(166,202),(167,203),(168,204)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BT | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | D8 | C4○D4 | C4○D8 | C7×D4 | C7×D8 | C7×C4○D4 | C7×C4○D8 |
| kernel | D8×C28 | C4×C56 | C7×D4⋊C4 | C7×C2.D8 | D4×C28 | C14×D8 | C7×D8 | C4×D8 | C4×C8 | D4⋊C4 | C2.D8 | C4×D4 | C2×D8 | D8 | C2×C28 | C28 | C28 | C14 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 4 | 2 | 4 | 12 | 24 | 12 | 24 |
Matrix representation of D8×C28 ►in GL3(𝔽113) generated by
| 15 | 0 | 0 |
| 0 | 109 | 0 |
| 0 | 0 | 109 |
| 1 | 0 | 0 |
| 0 | 62 | 31 |
| 0 | 51 | 0 |
| 112 | 0 | 0 |
| 0 | 0 | 31 |
| 0 | 62 | 0 |
G:=sub<GL(3,GF(113))| [15,0,0,0,109,0,0,0,109],[1,0,0,0,62,51,0,31,0],[112,0,0,0,0,62,0,31,0] >;
D8×C28 in GAP, Magma, Sage, TeX
D_8\times C_{28} % in TeX
G:=Group("D8xC28"); // GroupNames label
G:=SmallGroup(448,845);
// by ID
G=gap.SmallGroup(448,845);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,604,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c|a^28=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations