metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.4D8, C4.12D56, C56.81D4, Dic28.2C4, M5(2).4D7, C8.5(C4×D7), C56.2(C2×C4), (C2×C8).47D14, (C2×C4).10D28, (C2×C28).100D4, C7⋊2(C8.17D4), C8.38(C7⋊D4), (C2×C14).9SD16, C4.19(D14⋊C4), C56.C4.6C2, (C2×C56).51C22, (C2×Dic28).6C2, (C7×M5(2)).5C2, C28.43(C22⋊C4), C22.7(C56⋊C2), C2.10(C2.D56), C14.18(D4⋊C4), SmallGroup(448,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.4D8
G = < a,b,c | a28=1, b8=a14, c2=a21, bab-1=a15, cac-1=a13, cbc-1=a21b7 >
2C2
28C4
28C4
2C14
14Q8
14Q8
28C2×C4
28Q8
28C8
4Dic7
4Dic7
2C16
7Q16
7Q16
14M4(2)
14Q16
14C2×Q8
2C112
Smallest permutation representation of C28.4D8
►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 91 46 65 8 112 53 58 15 105 32 79 22 98 39 72)(2 106 47 80 9 99 54 73 16 92 33 66 23 85 40 59)(3 93 48 67 10 86 55 60 17 107 34 81 24 100 41 74)(4 108 49 82 11 101 56 75 18 94 35 68 25 87 42 61)(5 95 50 69 12 88 29 62 19 109 36 83 26 102 43 76)(6 110 51 84 13 103 30 77 20 96 37 70 27 89 44 63)(7 97 52 71 14 90 31 64 21 111 38 57 28 104 45 78)(113 177 163 215 120 170 142 208 127 191 149 201 134 184 156 222)(114 192 164 202 121 185 143 223 128 178 150 216 135 171 157 209)(115 179 165 217 122 172 144 210 129 193 151 203 136 186 158 224)(116 194 166 204 123 187 145 197 130 180 152 218 137 173 159 211)(117 181 167 219 124 174 146 212 131 195 153 205 138 188 160 198)(118 196 168 206 125 189 147 199 132 182 154 220 139 175 161 213)(119 183 141 221 126 176 148 214 133 169 155 207 140 190 162 200)
(1 222 22 215 15 208 8 201)(2 207 23 200 16 221 9 214)(3 220 24 213 17 206 10 199)(4 205 25 198 18 219 11 212)(5 218 26 211 19 204 12 197)(6 203 27 224 20 217 13 210)(7 216 28 209 21 202 14 223)(29 187 50 180 43 173 36 194)(30 172 51 193 44 186 37 179)(31 185 52 178 45 171 38 192)(32 170 53 191 46 184 39 177)(33 183 54 176 47 169 40 190)(34 196 55 189 48 182 41 175)(35 181 56 174 49 195 42 188)(57 121 78 114 71 135 64 128)(58 134 79 127 72 120 65 113)(59 119 80 140 73 133 66 126)(60 132 81 125 74 118 67 139)(61 117 82 138 75 131 68 124)(62 130 83 123 76 116 69 137)(63 115 84 136 77 129 70 122)(85 141 106 162 99 155 92 148)(86 154 107 147 100 168 93 161)(87 167 108 160 101 153 94 146)(88 152 109 145 102 166 95 159)(89 165 110 158 103 151 96 144)(90 150 111 143 104 164 97 157)(91 163 112 156 105 149 98 142)
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), (1,91,46,65,8,112,53,58,15,105,32,79,22,98,39,72)(2,106,47,80,9,99,54,73,16,92,33,66,23,85,40,59)(3,93,48,67,10,86,55,60,17,107,34,81,24,100,41,74)(4,108,49,82,11,101,56,75,18,94,35,68,25,87,42,61)(5,95,50,69,12,88,29,62,19,109,36,83,26,102,43,76)(6,110,51,84,13,103,30,77,20,96,37,70,27,89,44,63)(7,97,52,71,14,90,31,64,21,111,38,57,28,104,45,78)(113,177,163,215,120,170,142,208,127,191,149,201,134,184,156,222)(114,192,164,202,121,185,143,223,128,178,150,216,135,171,157,209)(115,179,165,217,122,172,144,210,129,193,151,203,136,186,158,224)(116,194,166,204,123,187,145,197,130,180,152,218,137,173,159,211)(117,181,167,219,124,174,146,212,131,195,153,205,138,188,160,198)(118,196,168,206,125,189,147,199,132,182,154,220,139,175,161,213)(119,183,141,221,126,176,148,214,133,169,155,207,140,190,162,200), (1,222,22,215,15,208,8,201)(2,207,23,200,16,221,9,214)(3,220,24,213,17,206,10,199)(4,205,25,198,18,219,11,212)(5,218,26,211,19,204,12,197)(6,203,27,224,20,217,13,210)(7,216,28,209,21,202,14,223)(29,187,50,180,43,173,36,194)(30,172,51,193,44,186,37,179)(31,185,52,178,45,171,38,192)(32,170,53,191,46,184,39,177)(33,183,54,176,47,169,40,190)(34,196,55,189,48,182,41,175)(35,181,56,174,49,195,42,188)(57,121,78,114,71,135,64,128)(58,134,79,127,72,120,65,113)(59,119,80,140,73,133,66,126)(60,132,81,125,74,118,67,139)(61,117,82,138,75,131,68,124)(62,130,83,123,76,116,69,137)(63,115,84,136,77,129,70,122)(85,141,106,162,99,155,92,148)(86,154,107,147,100,168,93,161)(87,167,108,160,101,153,94,146)(88,152,109,145,102,166,95,159)(89,165,110,158,103,151,96,144)(90,150,111,143,104,164,97,157)(91,163,112,156,105,149,98,142)>;
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), (1,91,46,65,8,112,53,58,15,105,32,79,22,98,39,72)(2,106,47,80,9,99,54,73,16,92,33,66,23,85,40,59)(3,93,48,67,10,86,55,60,17,107,34,81,24,100,41,74)(4,108,49,82,11,101,56,75,18,94,35,68,25,87,42,61)(5,95,50,69,12,88,29,62,19,109,36,83,26,102,43,76)(6,110,51,84,13,103,30,77,20,96,37,70,27,89,44,63)(7,97,52,71,14,90,31,64,21,111,38,57,28,104,45,78)(113,177,163,215,120,170,142,208,127,191,149,201,134,184,156,222)(114,192,164,202,121,185,143,223,128,178,150,216,135,171,157,209)(115,179,165,217,122,172,144,210,129,193,151,203,136,186,158,224)(116,194,166,204,123,187,145,197,130,180,152,218,137,173,159,211)(117,181,167,219,124,174,146,212,131,195,153,205,138,188,160,198)(118,196,168,206,125,189,147,199,132,182,154,220,139,175,161,213)(119,183,141,221,126,176,148,214,133,169,155,207,140,190,162,200), (1,222,22,215,15,208,8,201)(2,207,23,200,16,221,9,214)(3,220,24,213,17,206,10,199)(4,205,25,198,18,219,11,212)(5,218,26,211,19,204,12,197)(6,203,27,224,20,217,13,210)(7,216,28,209,21,202,14,223)(29,187,50,180,43,173,36,194)(30,172,51,193,44,186,37,179)(31,185,52,178,45,171,38,192)(32,170,53,191,46,184,39,177)(33,183,54,176,47,169,40,190)(34,196,55,189,48,182,41,175)(35,181,56,174,49,195,42,188)(57,121,78,114,71,135,64,128)(58,134,79,127,72,120,65,113)(59,119,80,140,73,133,66,126)(60,132,81,125,74,118,67,139)(61,117,82,138,75,131,68,124)(62,130,83,123,76,116,69,137)(63,115,84,136,77,129,70,122)(85,141,106,162,99,155,92,148)(86,154,107,147,100,168,93,161)(87,167,108,160,101,153,94,146)(88,152,109,145,102,166,95,159)(89,165,110,158,103,151,96,144)(90,150,111,143,104,164,97,157)(91,163,112,156,105,149,98,142) );
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)], [(1,91,46,65,8,112,53,58,15,105,32,79,22,98,39,72),(2,106,47,80,9,99,54,73,16,92,33,66,23,85,40,59),(3,93,48,67,10,86,55,60,17,107,34,81,24,100,41,74),(4,108,49,82,11,101,56,75,18,94,35,68,25,87,42,61),(5,95,50,69,12,88,29,62,19,109,36,83,26,102,43,76),(6,110,51,84,13,103,30,77,20,96,37,70,27,89,44,63),(7,97,52,71,14,90,31,64,21,111,38,57,28,104,45,78),(113,177,163,215,120,170,142,208,127,191,149,201,134,184,156,222),(114,192,164,202,121,185,143,223,128,178,150,216,135,171,157,209),(115,179,165,217,122,172,144,210,129,193,151,203,136,186,158,224),(116,194,166,204,123,187,145,197,130,180,152,218,137,173,159,211),(117,181,167,219,124,174,146,212,131,195,153,205,138,188,160,198),(118,196,168,206,125,189,147,199,132,182,154,220,139,175,161,213),(119,183,141,221,126,176,148,214,133,169,155,207,140,190,162,200)], [(1,222,22,215,15,208,8,201),(2,207,23,200,16,221,9,214),(3,220,24,213,17,206,10,199),(4,205,25,198,18,219,11,212),(5,218,26,211,19,204,12,197),(6,203,27,224,20,217,13,210),(7,216,28,209,21,202,14,223),(29,187,50,180,43,173,36,194),(30,172,51,193,44,186,37,179),(31,185,52,178,45,171,38,192),(32,170,53,191,46,184,39,177),(33,183,54,176,47,169,40,190),(34,196,55,189,48,182,41,175),(35,181,56,174,49,195,42,188),(57,121,78,114,71,135,64,128),(58,134,79,127,72,120,65,113),(59,119,80,140,73,133,66,126),(60,132,81,125,74,118,67,139),(61,117,82,138,75,131,68,124),(62,130,83,123,76,116,69,137),(63,115,84,136,77,129,70,122),(85,141,106,162,99,155,92,148),(86,154,107,147,100,168,93,161),(87,167,108,160,101,153,94,146),(88,152,109,145,102,166,95,159),(89,165,110,158,103,151,96,144),(90,150,111,143,104,164,97,157),(91,163,112,156,105,149,98,142)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | 14E | 14F | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56R | 112A | ··· | 112X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 4 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D8 | SD16 | D14 | C4×D7 | C7⋊D4 | D28 | D56 | C56⋊C2 | C8.17D4 | C28.4D8 |
| kernel | C28.4D8 | C56.C4 | C7×M5(2) | C2×Dic28 | Dic28 | C56 | C2×C28 | M5(2) | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 12 | 12 | 2 | 12 |
Matrix representation of C28.4D8 ►in GL6(𝔽113)
| 0 | 9 | 0 | 0 | 0 | 0 |
| 25 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 111 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 95 | 99 | 112 | 7 |
| 0 | 0 | 13 | 11 | 32 | 1 |
| 22 | 48 | 0 | 0 | 0 | 0 |
| 58 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 111 | 98 | 81 | 0 |
| 0 | 0 | 0 | 0 | 32 | 1 |
| 0 | 0 | 90 | 112 | 100 | 109 |
| 0 | 0 | 27 | 32 | 77 | 15 |
| 60 | 107 | 0 | 0 | 0 | 0 |
| 16 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 21 | 94 | 99 |
| 0 | 0 | 79 | 84 | 111 | 24 |
| 0 | 0 | 98 | 31 | 101 | 32 |
| 0 | 0 | 21 | 35 | 45 | 106 |
G:=sub<GL(6,GF(113))| [0,25,0,0,0,0,9,79,0,0,0,0,0,0,112,1,95,13,0,0,111,1,99,11,0,0,0,0,112,32,0,0,0,0,7,1],[22,58,0,0,0,0,48,29,0,0,0,0,0,0,111,0,90,27,0,0,98,0,112,32,0,0,81,32,100,77,0,0,0,1,109,15],[60,16,0,0,0,0,107,53,0,0,0,0,0,0,48,79,98,21,0,0,21,84,31,35,0,0,94,111,101,45,0,0,99,24,32,106] >;
C28.4D8 in GAP, Magma, Sage, TeX
C_{28}._4D_8 % in TeX
G:=Group("C28.4D8"); // GroupNames label
G:=SmallGroup(448,74);
// by ID
G=gap.SmallGroup(448,74);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,85,92,422,387,268,570,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^28=1,b^8=a^14,c^2=a^21,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=a^21*b^7>;
// generators/relations
Export