direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C28, C8⋊5(C2×C28), (C4×C8)⋊11C14, (C4×C56)⋊27C2, C56⋊27(C2×C4), (C4×Q8)⋊1C14, Q8⋊1(C2×C28), (Q8×C28)⋊21C2, (C4×D4).4C14, D4.1(C2×C28), C4.Q8⋊14C14, C2.13(D4×C28), (D4×C28).19C2, C14.115(C4×D4), (C2×C28).361D4, C2.4(C14×SD16), Q8⋊C4⋊21C14, D4⋊C4.8C14, C42.71(C2×C14), C4.10(C22×C28), (C2×SD16).5C14, C14.84(C2×SD16), C22.52(D4×C14), C14.117(C4○D8), C28.257(C4○D4), (C4×C28).356C22, (C2×C28).905C23, C28.155(C22×C4), (C2×C56).437C22, (C14×SD16).10C2, (D4×C14).291C22, (Q8×C14).254C22, C4.2(C7×C4○D4), C2.4(C7×C4○D8), (C7×Q8)⋊13(C2×C4), (C7×C4.Q8)⋊29C2, (C2×C4).51(C7×D4), C4⋊C4.46(C2×C14), (C2×C8).66(C2×C14), (C7×D4).18(C2×C4), (C7×Q8⋊C4)⋊44C2, (C2×D4).49(C2×C14), (C2×C14).628(C2×D4), (C2×Q8).39(C2×C14), (C7×D4⋊C4).17C2, (C7×C4⋊C4).367C22, (C2×C4).80(C22×C14), SmallGroup(448,846)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C28
G = < a,b,c | a28=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 202 in 122 conjugacy classes, 74 normal (50 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C28, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C4×SD16, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C22×C28, D4×C14, Q8×C14, C4×C56, C7×D4⋊C4, C7×Q8⋊C4, C7×C4.Q8, D4×C28, Q8×C28, C14×SD16, SD16×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, SD16, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C2×SD16, C4○D8, C2×C28, C7×D4, C22×C14, C4×SD16, C7×SD16, C22×C28, D4×C14, C7×C4○D4, D4×C28, C14×SD16, C7×C4○D8, SD16×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 77 180 90 126 41 146 212)(2 78 181 91 127 42 147 213)(3 79 182 92 128 43 148 214)(4 80 183 93 129 44 149 215)(5 81 184 94 130 45 150 216)(6 82 185 95 131 46 151 217)(7 83 186 96 132 47 152 218)(8 84 187 97 133 48 153 219)(9 57 188 98 134 49 154 220)(10 58 189 99 135 50 155 221)(11 59 190 100 136 51 156 222)(12 60 191 101 137 52 157 223)(13 61 192 102 138 53 158 224)(14 62 193 103 139 54 159 197)(15 63 194 104 140 55 160 198)(16 64 195 105 113 56 161 199)(17 65 196 106 114 29 162 200)(18 66 169 107 115 30 163 201)(19 67 170 108 116 31 164 202)(20 68 171 109 117 32 165 203)(21 69 172 110 118 33 166 204)(22 70 173 111 119 34 167 205)(23 71 174 112 120 35 168 206)(24 72 175 85 121 36 141 207)(25 73 176 86 122 37 142 208)(26 74 177 87 123 38 143 209)(27 75 178 88 124 39 144 210)(28 76 179 89 125 40 145 211)
(29 200)(30 201)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 217)(47 218)(48 219)(49 220)(50 221)(51 222)(52 223)(53 224)(54 197)(55 198)(56 199)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(141 175)(142 176)(143 177)(144 178)(145 179)(146 180)(147 181)(148 182)(149 183)(150 184)(151 185)(152 186)(153 187)(154 188)(155 189)(156 190)(157 191)(158 192)(159 193)(160 194)(161 195)(162 196)(163 169)(164 170)(165 171)(166 172)(167 173)(168 174)
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), (1,77,180,90,126,41,146,212)(2,78,181,91,127,42,147,213)(3,79,182,92,128,43,148,214)(4,80,183,93,129,44,149,215)(5,81,184,94,130,45,150,216)(6,82,185,95,131,46,151,217)(7,83,186,96,132,47,152,218)(8,84,187,97,133,48,153,219)(9,57,188,98,134,49,154,220)(10,58,189,99,135,50,155,221)(11,59,190,100,136,51,156,222)(12,60,191,101,137,52,157,223)(13,61,192,102,138,53,158,224)(14,62,193,103,139,54,159,197)(15,63,194,104,140,55,160,198)(16,64,195,105,113,56,161,199)(17,65,196,106,114,29,162,200)(18,66,169,107,115,30,163,201)(19,67,170,108,116,31,164,202)(20,68,171,109,117,32,165,203)(21,69,172,110,118,33,166,204)(22,70,173,111,119,34,167,205)(23,71,174,112,120,35,168,206)(24,72,175,85,121,36,141,207)(25,73,176,86,122,37,142,208)(26,74,177,87,123,38,143,209)(27,75,178,88,124,39,144,210)(28,76,179,89,125,40,145,211), (29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,197)(55,198)(56,199)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,181)(148,182)(149,183)(150,184)(151,185)(152,186)(153,187)(154,188)(155,189)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174)>;
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), (1,77,180,90,126,41,146,212)(2,78,181,91,127,42,147,213)(3,79,182,92,128,43,148,214)(4,80,183,93,129,44,149,215)(5,81,184,94,130,45,150,216)(6,82,185,95,131,46,151,217)(7,83,186,96,132,47,152,218)(8,84,187,97,133,48,153,219)(9,57,188,98,134,49,154,220)(10,58,189,99,135,50,155,221)(11,59,190,100,136,51,156,222)(12,60,191,101,137,52,157,223)(13,61,192,102,138,53,158,224)(14,62,193,103,139,54,159,197)(15,63,194,104,140,55,160,198)(16,64,195,105,113,56,161,199)(17,65,196,106,114,29,162,200)(18,66,169,107,115,30,163,201)(19,67,170,108,116,31,164,202)(20,68,171,109,117,32,165,203)(21,69,172,110,118,33,166,204)(22,70,173,111,119,34,167,205)(23,71,174,112,120,35,168,206)(24,72,175,85,121,36,141,207)(25,73,176,86,122,37,142,208)(26,74,177,87,123,38,143,209)(27,75,178,88,124,39,144,210)(28,76,179,89,125,40,145,211), (29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,197)(55,198)(56,199)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,181)(148,182)(149,183)(150,184)(151,185)(152,186)(153,187)(154,188)(155,189)(156,190)(157,191)(158,192)(159,193)(160,194)(161,195)(162,196)(163,169)(164,170)(165,171)(166,172)(167,173)(168,174) );
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)], [(1,77,180,90,126,41,146,212),(2,78,181,91,127,42,147,213),(3,79,182,92,128,43,148,214),(4,80,183,93,129,44,149,215),(5,81,184,94,130,45,150,216),(6,82,185,95,131,46,151,217),(7,83,186,96,132,47,152,218),(8,84,187,97,133,48,153,219),(9,57,188,98,134,49,154,220),(10,58,189,99,135,50,155,221),(11,59,190,100,136,51,156,222),(12,60,191,101,137,52,157,223),(13,61,192,102,138,53,158,224),(14,62,193,103,139,54,159,197),(15,63,194,104,140,55,160,198),(16,64,195,105,113,56,161,199),(17,65,196,106,114,29,162,200),(18,66,169,107,115,30,163,201),(19,67,170,108,116,31,164,202),(20,68,171,109,117,32,165,203),(21,69,172,110,118,33,166,204),(22,70,173,111,119,34,167,205),(23,71,174,112,120,35,168,206),(24,72,175,85,121,36,141,207),(25,73,176,86,122,37,142,208),(26,74,177,87,123,38,143,209),(27,75,178,88,124,39,144,210),(28,76,179,89,125,40,145,211)], [(29,200),(30,201),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,217),(47,218),(48,219),(49,220),(50,221),(51,222),(52,223),(53,224),(54,197),(55,198),(56,199),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(141,175),(142,176),(143,177),(144,178),(145,179),(146,180),(147,181),(148,182),(149,183),(150,184),(151,185),(152,186),(153,187),(154,188),(155,189),(156,190),(157,191),(158,192),(159,193),(160,194),(161,195),(162,196),(163,169),(164,170),(165,171),(166,172),(167,173),(168,174)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28CF | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | SD16 | C4○D4 | C4○D8 | C7×D4 | C7×SD16 | C7×C4○D4 | C7×C4○D8 |
| kernel | SD16×C28 | C4×C56 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C4.Q8 | D4×C28 | Q8×C28 | C14×SD16 | C7×SD16 | C4×SD16 | C4×C8 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C4×D4 | C4×Q8 | C2×SD16 | SD16 | C2×C28 | C28 | C28 | C14 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 48 | 2 | 4 | 2 | 4 | 12 | 24 | 12 | 24 |
Matrix representation of SD16×C28 ►in GL3(𝔽113) generated by
| 15 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 26 | 100 |
| 0 | 26 | 0 |
| 112 | 0 | 0 |
| 0 | 1 | 112 |
| 0 | 0 | 112 |
G:=sub<GL(3,GF(113))| [15,0,0,0,64,0,0,0,64],[1,0,0,0,26,26,0,100,0],[112,0,0,0,1,0,0,112,112] >;
SD16×C28 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{28} % in TeX
G:=Group("SD16xC28"); // GroupNames label
G:=SmallGroup(448,846);
// by ID
G=gap.SmallGroup(448,846);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,604,9804,4911,172]);
// 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^3>;
// generators/relations