metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.58D8, C56.41D4, D8.1Dic7, Q16.1Dic7, C4○D8.1D7, (C7×D8).1C4, C56.15(C2×C4), (C7×Q16).1C4, C56.C4⋊5C2, C8.9(C2×Dic7), C4.31(D4⋊D7), (C2×C28).119D4, (C2×C8).250D14, C7⋊3(D8.C4), C8.31(C7⋊D4), (C2×C56).38C22, (C2×C14).12SD16, C4.6(C23.D7), C28.18(C22⋊C4), C22.1(D4.D7), C14.31(D4⋊C4), C2.11(D4⋊Dic7), (C2×C7⋊C16)⋊2C2, (C7×C4○D8).1C2, (C2×C4).121(C7⋊D4), SmallGroup(448,124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.58D8
G = < a,b,c | a28=1, b8=a14, c2=a21, bab-1=cac-1=a13, cbc-1=a7b7 >
Smallest permutation representation of C28.58D8
►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)
(1 205 42 177 22 198 35 170 15 219 56 191 8 212 49 184)(2 218 43 190 23 211 36 183 16 204 29 176 9 197 50 169)(3 203 44 175 24 224 37 196 17 217 30 189 10 210 51 182)(4 216 45 188 25 209 38 181 18 202 31 174 11 223 52 195)(5 201 46 173 26 222 39 194 19 215 32 187 12 208 53 180)(6 214 47 186 27 207 40 179 20 200 33 172 13 221 54 193)(7 199 48 171 28 220 41 192 21 213 34 185 14 206 55 178)(57 166 96 137 64 145 103 116 71 152 110 123 78 159 89 130)(58 151 97 122 65 158 104 129 72 165 111 136 79 144 90 115)(59 164 98 135 66 143 105 114 73 150 112 121 80 157 91 128)(60 149 99 120 67 156 106 127 74 163 85 134 81 142 92 113)(61 162 100 133 68 141 107 140 75 148 86 119 82 155 93 126)(62 147 101 118 69 154 108 125 76 161 87 132 83 168 94 139)(63 160 102 131 70 167 109 138 77 146 88 117 84 153 95 124)
(1 124 22 117 15 138 8 131)(2 137 23 130 16 123 9 116)(3 122 24 115 17 136 10 129)(4 135 25 128 18 121 11 114)(5 120 26 113 19 134 12 127)(6 133 27 126 20 119 13 140)(7 118 28 139 21 132 14 125)(29 166 50 159 43 152 36 145)(30 151 51 144 44 165 37 158)(31 164 52 157 45 150 38 143)(32 149 53 142 46 163 39 156)(33 162 54 155 47 148 40 141)(34 147 55 168 48 161 41 154)(35 160 56 153 49 146 42 167)(57 176 78 169 71 190 64 183)(58 189 79 182 72 175 65 196)(59 174 80 195 73 188 66 181)(60 187 81 180 74 173 67 194)(61 172 82 193 75 186 68 179)(62 185 83 178 76 171 69 192)(63 170 84 191 77 184 70 177)(85 215 106 208 99 201 92 222)(86 200 107 221 100 214 93 207)(87 213 108 206 101 199 94 220)(88 198 109 219 102 212 95 205)(89 211 110 204 103 197 96 218)(90 224 111 217 104 210 97 203)(91 209 112 202 105 223 98 216)
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), (1,205,42,177,22,198,35,170,15,219,56,191,8,212,49,184)(2,218,43,190,23,211,36,183,16,204,29,176,9,197,50,169)(3,203,44,175,24,224,37,196,17,217,30,189,10,210,51,182)(4,216,45,188,25,209,38,181,18,202,31,174,11,223,52,195)(5,201,46,173,26,222,39,194,19,215,32,187,12,208,53,180)(6,214,47,186,27,207,40,179,20,200,33,172,13,221,54,193)(7,199,48,171,28,220,41,192,21,213,34,185,14,206,55,178)(57,166,96,137,64,145,103,116,71,152,110,123,78,159,89,130)(58,151,97,122,65,158,104,129,72,165,111,136,79,144,90,115)(59,164,98,135,66,143,105,114,73,150,112,121,80,157,91,128)(60,149,99,120,67,156,106,127,74,163,85,134,81,142,92,113)(61,162,100,133,68,141,107,140,75,148,86,119,82,155,93,126)(62,147,101,118,69,154,108,125,76,161,87,132,83,168,94,139)(63,160,102,131,70,167,109,138,77,146,88,117,84,153,95,124), (1,124,22,117,15,138,8,131)(2,137,23,130,16,123,9,116)(3,122,24,115,17,136,10,129)(4,135,25,128,18,121,11,114)(5,120,26,113,19,134,12,127)(6,133,27,126,20,119,13,140)(7,118,28,139,21,132,14,125)(29,166,50,159,43,152,36,145)(30,151,51,144,44,165,37,158)(31,164,52,157,45,150,38,143)(32,149,53,142,46,163,39,156)(33,162,54,155,47,148,40,141)(34,147,55,168,48,161,41,154)(35,160,56,153,49,146,42,167)(57,176,78,169,71,190,64,183)(58,189,79,182,72,175,65,196)(59,174,80,195,73,188,66,181)(60,187,81,180,74,173,67,194)(61,172,82,193,75,186,68,179)(62,185,83,178,76,171,69,192)(63,170,84,191,77,184,70,177)(85,215,106,208,99,201,92,222)(86,200,107,221,100,214,93,207)(87,213,108,206,101,199,94,220)(88,198,109,219,102,212,95,205)(89,211,110,204,103,197,96,218)(90,224,111,217,104,210,97,203)(91,209,112,202,105,223,98,216)>;
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), (1,205,42,177,22,198,35,170,15,219,56,191,8,212,49,184)(2,218,43,190,23,211,36,183,16,204,29,176,9,197,50,169)(3,203,44,175,24,224,37,196,17,217,30,189,10,210,51,182)(4,216,45,188,25,209,38,181,18,202,31,174,11,223,52,195)(5,201,46,173,26,222,39,194,19,215,32,187,12,208,53,180)(6,214,47,186,27,207,40,179,20,200,33,172,13,221,54,193)(7,199,48,171,28,220,41,192,21,213,34,185,14,206,55,178)(57,166,96,137,64,145,103,116,71,152,110,123,78,159,89,130)(58,151,97,122,65,158,104,129,72,165,111,136,79,144,90,115)(59,164,98,135,66,143,105,114,73,150,112,121,80,157,91,128)(60,149,99,120,67,156,106,127,74,163,85,134,81,142,92,113)(61,162,100,133,68,141,107,140,75,148,86,119,82,155,93,126)(62,147,101,118,69,154,108,125,76,161,87,132,83,168,94,139)(63,160,102,131,70,167,109,138,77,146,88,117,84,153,95,124), (1,124,22,117,15,138,8,131)(2,137,23,130,16,123,9,116)(3,122,24,115,17,136,10,129)(4,135,25,128,18,121,11,114)(5,120,26,113,19,134,12,127)(6,133,27,126,20,119,13,140)(7,118,28,139,21,132,14,125)(29,166,50,159,43,152,36,145)(30,151,51,144,44,165,37,158)(31,164,52,157,45,150,38,143)(32,149,53,142,46,163,39,156)(33,162,54,155,47,148,40,141)(34,147,55,168,48,161,41,154)(35,160,56,153,49,146,42,167)(57,176,78,169,71,190,64,183)(58,189,79,182,72,175,65,196)(59,174,80,195,73,188,66,181)(60,187,81,180,74,173,67,194)(61,172,82,193,75,186,68,179)(62,185,83,178,76,171,69,192)(63,170,84,191,77,184,70,177)(85,215,106,208,99,201,92,222)(86,200,107,221,100,214,93,207)(87,213,108,206,101,199,94,220)(88,198,109,219,102,212,95,205)(89,211,110,204,103,197,96,218)(90,224,111,217,104,210,97,203)(91,209,112,202,105,223,98,216) );
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)], [(1,205,42,177,22,198,35,170,15,219,56,191,8,212,49,184),(2,218,43,190,23,211,36,183,16,204,29,176,9,197,50,169),(3,203,44,175,24,224,37,196,17,217,30,189,10,210,51,182),(4,216,45,188,25,209,38,181,18,202,31,174,11,223,52,195),(5,201,46,173,26,222,39,194,19,215,32,187,12,208,53,180),(6,214,47,186,27,207,40,179,20,200,33,172,13,221,54,193),(7,199,48,171,28,220,41,192,21,213,34,185,14,206,55,178),(57,166,96,137,64,145,103,116,71,152,110,123,78,159,89,130),(58,151,97,122,65,158,104,129,72,165,111,136,79,144,90,115),(59,164,98,135,66,143,105,114,73,150,112,121,80,157,91,128),(60,149,99,120,67,156,106,127,74,163,85,134,81,142,92,113),(61,162,100,133,68,141,107,140,75,148,86,119,82,155,93,126),(62,147,101,118,69,154,108,125,76,161,87,132,83,168,94,139),(63,160,102,131,70,167,109,138,77,146,88,117,84,153,95,124)], [(1,124,22,117,15,138,8,131),(2,137,23,130,16,123,9,116),(3,122,24,115,17,136,10,129),(4,135,25,128,18,121,11,114),(5,120,26,113,19,134,12,127),(6,133,27,126,20,119,13,140),(7,118,28,139,21,132,14,125),(29,166,50,159,43,152,36,145),(30,151,51,144,44,165,37,158),(31,164,52,157,45,150,38,143),(32,149,53,142,46,163,39,156),(33,162,54,155,47,148,40,141),(34,147,55,168,48,161,41,154),(35,160,56,153,49,146,42,167),(57,176,78,169,71,190,64,183),(58,189,79,182,72,175,65,196),(59,174,80,195,73,188,66,181),(60,187,81,180,74,173,67,194),(61,172,82,193,75,186,68,179),(62,185,83,178,76,171,69,192),(63,170,84,191,77,184,70,177),(85,215,106,208,99,201,92,222),(86,200,107,221,100,214,93,207),(87,213,108,206,101,199,94,220),(88,198,109,219,102,212,95,205),(89,211,110,204,103,197,96,218),(90,224,111,217,104,210,97,203),(91,209,112,202,105,223,98,216)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 16A | ··· | 16H | 28A | ··· | 28F | 28G | 28H | 28I | 28J | ··· | 28O | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | Dic7 | Dic7 | C7⋊D4 | C7⋊D4 | D8.C4 | D4⋊D7 | D4.D7 | C28.58D8 |
| kernel | C28.58D8 | C2×C7⋊C16 | C56.C4 | C7×C4○D8 | C7×D8 | C7×Q16 | C56 | C2×C28 | C4○D8 | C28 | C2×C14 | C2×C8 | D8 | Q16 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 3 | 2 | 2 | 3 | 3 | 3 | 6 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of C28.58D8 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 34 | 59 |
| 0 | 0 | 89 | 88 |
| 71 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 10 | 23 |
| 0 | 0 | 30 | 103 |
| 0 | 40 | 0 | 0 |
| 71 | 0 | 0 | 0 |
| 0 | 0 | 42 | 47 |
| 0 | 0 | 25 | 71 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,34,89,0,0,59,88],[71,0,0,0,0,40,0,0,0,0,10,30,0,0,23,103],[0,71,0,0,40,0,0,0,0,0,42,25,0,0,47,71] >;
C28.58D8 in GAP, Magma, Sage, TeX
C_{28}._{58}D_8 % in TeX
G:=Group("C28.58D8"); // GroupNames label
G:=SmallGroup(448,124);
// by ID
G=gap.SmallGroup(448,124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,184,675,346,192,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^8=a^14,c^2=a^21,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=a^7*b^7>;
// generators/relations
Export