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## G = C28.10C42order 448 = 26·7

### 3rd non-split extension by C28 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.10C42
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C22×C28 — C2×C4.Dic7 — C28.10C42
 Lower central C7 — C14 — C28 — C28.10C42
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C28.10C42
G = < a,b,c | a28=1, b4=c4=a14, bab-1=a-1, ac=ca, cbc-1=a7b >

Subgroups: 260 in 90 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C22×C8, C2×M4(2), C7⋊C8, C56, C2×C28, C2×C28, C22×C14, C4.C42, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C2×C56, C22×C28, C2×C4.Dic7, C22×C56, C28.10C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C8.C4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C56.C4, C14.C42, C28.10C42

Smallest permutation representation of C28.10C42
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 54 8 47 15 40 22 33)(2 53 9 46 16 39 23 32)(3 52 10 45 17 38 24 31)(4 51 11 44 18 37 25 30)(5 50 12 43 19 36 26 29)(6 49 13 42 20 35 27 56)(7 48 14 41 21 34 28 55)(57 128 78 135 71 114 64 121)(58 127 79 134 72 113 65 120)(59 126 80 133 73 140 66 119)(60 125 81 132 74 139 67 118)(61 124 82 131 75 138 68 117)(62 123 83 130 76 137 69 116)(63 122 84 129 77 136 70 115)(85 159 92 152 99 145 106 166)(86 158 93 151 100 144 107 165)(87 157 94 150 101 143 108 164)(88 156 95 149 102 142 109 163)(89 155 96 148 103 141 110 162)(90 154 97 147 104 168 111 161)(91 153 98 146 105 167 112 160)(169 199 190 206 183 213 176 220)(170 198 191 205 184 212 177 219)(171 197 192 204 185 211 178 218)(172 224 193 203 186 210 179 217)(173 223 194 202 187 209 180 216)(174 222 195 201 188 208 181 215)(175 221 196 200 189 207 182 214)
(1 98 205 123 15 112 219 137)(2 99 206 124 16 85 220 138)(3 100 207 125 17 86 221 139)(4 101 208 126 18 87 222 140)(5 102 209 127 19 88 223 113)(6 103 210 128 20 89 224 114)(7 104 211 129 21 90 197 115)(8 105 212 130 22 91 198 116)(9 106 213 131 23 92 199 117)(10 107 214 132 24 93 200 118)(11 108 215 133 25 94 201 119)(12 109 216 134 26 95 202 120)(13 110 217 135 27 96 203 121)(14 111 218 136 28 97 204 122)(29 156 173 79 43 142 187 65)(30 157 174 80 44 143 188 66)(31 158 175 81 45 144 189 67)(32 159 176 82 46 145 190 68)(33 160 177 83 47 146 191 69)(34 161 178 84 48 147 192 70)(35 162 179 57 49 148 193 71)(36 163 180 58 50 149 194 72)(37 164 181 59 51 150 195 73)(38 165 182 60 52 151 196 74)(39 166 183 61 53 152 169 75)(40 167 184 62 54 153 170 76)(41 168 185 63 55 154 171 77)(42 141 186 64 56 155 172 78)```

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 2 ··· 2 28 ··· 28 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - image C1 C2 C2 C4 C4 D4 Q8 D7 Dic7 D14 C8.C4 C4×D7 D28 C7⋊D4 Dic14 C56.C4 kernel C28.10C42 C2×C4.Dic7 C22×C56 C4.Dic7 C2×C56 C2×C28 C22×C14 C22×C8 C2×C8 C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 1 8 4 3 1 3 6 3 8 12 6 12 6 48

Matrix representation of C28.10C42 in GL3(𝔽113) generated by

 1 0 0 0 53 0 0 0 32
,
 98 0 0 0 0 1 0 98 0
,
 15 0 0 0 18 0 0 0 44
`G:=sub<GL(3,GF(113))| [1,0,0,0,53,0,0,0,32],[98,0,0,0,0,98,0,1,0],[15,0,0,0,18,0,0,0,44] >;`

C28.10C42 in GAP, Magma, Sage, TeX

`C_{28}._{10}C_4^2`
`% in TeX`

`G:=Group("C28.10C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,109);`
`// by ID`

`G=gap.SmallGroup(448,109);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,184,1123,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=1,b^4=c^4=a^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;`
`// generators/relations`

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