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## G = C28.57D8order 448 = 26·7

### 11st non-split extension by C28 of D8 acting via D8/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.57D8
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×C28 — C28⋊C8 — C28.57D8
 Lower central C7 — C14 — C28 — C28.57D8
 Upper central C1 — C2×C4 — C42 — C4×D4

Generators and relations for C28.57D8
G = < a,b,c | a28=b8=1, c2=a21, bab-1=cac-1=a13, cbc-1=a21b-1 >

Subgroups: 244 in 82 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C4×C8, C4⋊C8, C4×D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4⋊C8, C2×C7⋊C8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C4×C7⋊C8, C28⋊C8, D4×C28, C28.57D8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D8, SD16, Dic7, D14, C22⋊C8, D4⋊C4, C4≀C2, C7⋊C8, C2×Dic7, C7⋊D4, D4⋊C8, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, C23.D7, C28.55D4, D4⋊Dic7, D42Dic7, C28.57D8

Smallest permutation representation of C28.57D8
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)
(1 166 99 196 124 199 44 77)(2 151 100 181 125 212 45 62)(3 164 101 194 126 197 46 75)(4 149 102 179 127 210 47 60)(5 162 103 192 128 223 48 73)(6 147 104 177 129 208 49 58)(7 160 105 190 130 221 50 71)(8 145 106 175 131 206 51 84)(9 158 107 188 132 219 52 69)(10 143 108 173 133 204 53 82)(11 156 109 186 134 217 54 67)(12 141 110 171 135 202 55 80)(13 154 111 184 136 215 56 65)(14 167 112 169 137 200 29 78)(15 152 85 182 138 213 30 63)(16 165 86 195 139 198 31 76)(17 150 87 180 140 211 32 61)(18 163 88 193 113 224 33 74)(19 148 89 178 114 209 34 59)(20 161 90 191 115 222 35 72)(21 146 91 176 116 207 36 57)(22 159 92 189 117 220 37 70)(23 144 93 174 118 205 38 83)(24 157 94 187 119 218 39 68)(25 142 95 172 120 203 40 81)(26 155 96 185 121 216 41 66)(27 168 97 170 122 201 42 79)(28 153 98 183 123 214 43 64)
(1 84 22 77 15 70 8 63)(2 69 23 62 16 83 9 76)(3 82 24 75 17 68 10 61)(4 67 25 60 18 81 11 74)(5 80 26 73 19 66 12 59)(6 65 27 58 20 79 13 72)(7 78 28 71 21 64 14 57)(29 160 50 153 43 146 36 167)(30 145 51 166 44 159 37 152)(31 158 52 151 45 144 38 165)(32 143 53 164 46 157 39 150)(33 156 54 149 47 142 40 163)(34 141 55 162 48 155 41 148)(35 154 56 147 49 168 42 161)(85 206 106 199 99 220 92 213)(86 219 107 212 100 205 93 198)(87 204 108 197 101 218 94 211)(88 217 109 210 102 203 95 224)(89 202 110 223 103 216 96 209)(90 215 111 208 104 201 97 222)(91 200 112 221 105 214 98 207)(113 172 134 193 127 186 120 179)(114 185 135 178 128 171 121 192)(115 170 136 191 129 184 122 177)(116 183 137 176 130 169 123 190)(117 196 138 189 131 182 124 175)(118 181 139 174 132 195 125 188)(119 194 140 187 133 180 126 173)```

`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), (1,166,99,196,124,199,44,77)(2,151,100,181,125,212,45,62)(3,164,101,194,126,197,46,75)(4,149,102,179,127,210,47,60)(5,162,103,192,128,223,48,73)(6,147,104,177,129,208,49,58)(7,160,105,190,130,221,50,71)(8,145,106,175,131,206,51,84)(9,158,107,188,132,219,52,69)(10,143,108,173,133,204,53,82)(11,156,109,186,134,217,54,67)(12,141,110,171,135,202,55,80)(13,154,111,184,136,215,56,65)(14,167,112,169,137,200,29,78)(15,152,85,182,138,213,30,63)(16,165,86,195,139,198,31,76)(17,150,87,180,140,211,32,61)(18,163,88,193,113,224,33,74)(19,148,89,178,114,209,34,59)(20,161,90,191,115,222,35,72)(21,146,91,176,116,207,36,57)(22,159,92,189,117,220,37,70)(23,144,93,174,118,205,38,83)(24,157,94,187,119,218,39,68)(25,142,95,172,120,203,40,81)(26,155,96,185,121,216,41,66)(27,168,97,170,122,201,42,79)(28,153,98,183,123,214,43,64), (1,84,22,77,15,70,8,63)(2,69,23,62,16,83,9,76)(3,82,24,75,17,68,10,61)(4,67,25,60,18,81,11,74)(5,80,26,73,19,66,12,59)(6,65,27,58,20,79,13,72)(7,78,28,71,21,64,14,57)(29,160,50,153,43,146,36,167)(30,145,51,166,44,159,37,152)(31,158,52,151,45,144,38,165)(32,143,53,164,46,157,39,150)(33,156,54,149,47,142,40,163)(34,141,55,162,48,155,41,148)(35,154,56,147,49,168,42,161)(85,206,106,199,99,220,92,213)(86,219,107,212,100,205,93,198)(87,204,108,197,101,218,94,211)(88,217,109,210,102,203,95,224)(89,202,110,223,103,216,96,209)(90,215,111,208,104,201,97,222)(91,200,112,221,105,214,98,207)(113,172,134,193,127,186,120,179)(114,185,135,178,128,171,121,192)(115,170,136,191,129,184,122,177)(116,183,137,176,130,169,123,190)(117,196,138,189,131,182,124,175)(118,181,139,174,132,195,125,188)(119,194,140,187,133,180,126,173)>;`

`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), (1,166,99,196,124,199,44,77)(2,151,100,181,125,212,45,62)(3,164,101,194,126,197,46,75)(4,149,102,179,127,210,47,60)(5,162,103,192,128,223,48,73)(6,147,104,177,129,208,49,58)(7,160,105,190,130,221,50,71)(8,145,106,175,131,206,51,84)(9,158,107,188,132,219,52,69)(10,143,108,173,133,204,53,82)(11,156,109,186,134,217,54,67)(12,141,110,171,135,202,55,80)(13,154,111,184,136,215,56,65)(14,167,112,169,137,200,29,78)(15,152,85,182,138,213,30,63)(16,165,86,195,139,198,31,76)(17,150,87,180,140,211,32,61)(18,163,88,193,113,224,33,74)(19,148,89,178,114,209,34,59)(20,161,90,191,115,222,35,72)(21,146,91,176,116,207,36,57)(22,159,92,189,117,220,37,70)(23,144,93,174,118,205,38,83)(24,157,94,187,119,218,39,68)(25,142,95,172,120,203,40,81)(26,155,96,185,121,216,41,66)(27,168,97,170,122,201,42,79)(28,153,98,183,123,214,43,64), (1,84,22,77,15,70,8,63)(2,69,23,62,16,83,9,76)(3,82,24,75,17,68,10,61)(4,67,25,60,18,81,11,74)(5,80,26,73,19,66,12,59)(6,65,27,58,20,79,13,72)(7,78,28,71,21,64,14,57)(29,160,50,153,43,146,36,167)(30,145,51,166,44,159,37,152)(31,158,52,151,45,144,38,165)(32,143,53,164,46,157,39,150)(33,156,54,149,47,142,40,163)(34,141,55,162,48,155,41,148)(35,154,56,147,49,168,42,161)(85,206,106,199,99,220,92,213)(86,219,107,212,100,205,93,198)(87,204,108,197,101,218,94,211)(88,217,109,210,102,203,95,224)(89,202,110,223,103,216,96,209)(90,215,111,208,104,201,97,222)(91,200,112,221,105,214,98,207)(113,172,134,193,127,186,120,179)(114,185,135,178,128,171,121,192)(115,170,136,191,129,184,122,177)(116,183,137,176,130,169,123,190)(117,196,138,189,131,182,124,175)(118,181,139,174,132,195,125,188)(119,194,140,187,133,180,126,173) );`

`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)], [(1,166,99,196,124,199,44,77),(2,151,100,181,125,212,45,62),(3,164,101,194,126,197,46,75),(4,149,102,179,127,210,47,60),(5,162,103,192,128,223,48,73),(6,147,104,177,129,208,49,58),(7,160,105,190,130,221,50,71),(8,145,106,175,131,206,51,84),(9,158,107,188,132,219,52,69),(10,143,108,173,133,204,53,82),(11,156,109,186,134,217,54,67),(12,141,110,171,135,202,55,80),(13,154,111,184,136,215,56,65),(14,167,112,169,137,200,29,78),(15,152,85,182,138,213,30,63),(16,165,86,195,139,198,31,76),(17,150,87,180,140,211,32,61),(18,163,88,193,113,224,33,74),(19,148,89,178,114,209,34,59),(20,161,90,191,115,222,35,72),(21,146,91,176,116,207,36,57),(22,159,92,189,117,220,37,70),(23,144,93,174,118,205,38,83),(24,157,94,187,119,218,39,68),(25,142,95,172,120,203,40,81),(26,155,96,185,121,216,41,66),(27,168,97,170,122,201,42,79),(28,153,98,183,123,214,43,64)], [(1,84,22,77,15,70,8,63),(2,69,23,62,16,83,9,76),(3,82,24,75,17,68,10,61),(4,67,25,60,18,81,11,74),(5,80,26,73,19,66,12,59),(6,65,27,58,20,79,13,72),(7,78,28,71,21,64,14,57),(29,160,50,153,43,146,36,167),(30,145,51,166,44,159,37,152),(31,158,52,151,45,144,38,165),(32,143,53,164,46,157,39,150),(33,156,54,149,47,142,40,163),(34,141,55,162,48,155,41,148),(35,154,56,147,49,168,42,161),(85,206,106,199,99,220,92,213),(86,219,107,212,100,205,93,198),(87,204,108,197,101,218,94,211),(88,217,109,210,102,203,95,224),(89,202,110,223,103,216,96,209),(90,215,111,208,104,201,97,222),(91,200,112,221,105,214,98,207),(113,172,134,193,127,186,120,179),(114,185,135,178,128,171,121,192),(115,170,136,191,129,184,122,177),(116,183,137,176,130,169,123,190),(117,196,138,189,131,182,124,175),(118,181,139,174,132,195,125,188),(119,194,140,187,133,180,126,173)]])`

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 14A ··· 14I 14J ··· 14U 28A ··· 28L 28M ··· 28AJ order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 1 1 1 1 2 2 2 2 4 4 2 2 2 14 ··· 14 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - - + - image C1 C2 C2 C2 C4 C4 C8 D4 D7 M4(2) D8 SD16 D14 Dic7 Dic7 C4≀C2 C7⋊D4 C7⋊C8 C4.Dic7 D4⋊D7 D4.D7 D4⋊2Dic7 kernel C28.57D8 C4×C7⋊C8 C28⋊C8 D4×C28 C7×C4⋊C4 D4×C14 C7×D4 C2×C28 C4×D4 C28 C28 C28 C42 C4⋊C4 C2×D4 C14 C2×C4 D4 C4 C4 C4 C2 # reps 1 1 1 1 2 2 8 2 3 2 2 2 3 3 3 4 12 12 12 3 3 6

Matrix representation of C28.57D8 in GL6(𝔽113)

 7 0 0 0 0 0 46 97 0 0 0 0 0 0 98 0 0 0 0 0 0 98 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 82 72 0 0 0 0 40 31 0 0 0 0 0 0 67 55 0 0 0 0 7 46 0 0 0 0 0 0 0 6 0 0 0 0 54 16
,
 82 72 0 0 0 0 40 31 0 0 0 0 0 0 46 58 0 0 0 0 86 67 0 0 0 0 0 0 0 107 0 0 0 0 54 0

`G:=sub<GL(6,GF(113))| [7,46,0,0,0,0,0,97,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[82,40,0,0,0,0,72,31,0,0,0,0,0,0,67,7,0,0,0,0,55,46,0,0,0,0,0,0,0,54,0,0,0,0,6,16],[82,40,0,0,0,0,72,31,0,0,0,0,0,0,46,86,0,0,0,0,58,67,0,0,0,0,0,0,0,54,0,0,0,0,107,0] >;`

C28.57D8 in GAP, Magma, Sage, TeX

`C_{28}._{57}D_8`
`% in TeX`

`G:=Group("C28.57D8");`
`// GroupNames label`

`G:=SmallGroup(448,91);`
`// by ID`

`G=gap.SmallGroup(448,91);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,100,1123,570,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^8=1,c^2=a^21,b*a*b^-1=c*a*c^-1=a^13,c*b*c^-1=a^21*b^-1>;`
`// generators/relations`

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