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## G = C28.12C42order 448 = 26·7

### 5th non-split extension by C28 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C28.12C42
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C4×Dic7 — C23.21D14 — C28.12C42
 Lower central C7 — C14 — C28.12C42
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C28.12C42
G = < a,b,c | a28=b4=1, c4=a14, bab-1=a-1, ac=ca, bc=cb >

Subgroups: 356 in 130 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C82M4(2), C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C2×C56, C22×C28, C8×Dic7, C56⋊C4, C2×C4.Dic7, C23.21D14, C22×C56, C28.12C42
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, C22×C4, Dic7, D14, C2×C42, C8○D4, C4×D7, C2×Dic7, C22×D7, C82M4(2), C4×Dic7, C2×C4×D7, C22×Dic7, D28.2C4, C2×C4×Dic7, C28.12C42

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

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 14A ··· 14U 28A ··· 28X 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 7 7 7 8 ··· 8 8 8 8 8 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 14 ··· 14 2 2 2 1 ··· 1 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 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 C4 C4 C4 C4 D7 Dic7 D14 D14 C8○D4 C4×D7 C4×D7 D28.2C4 kernel C28.12C42 C8×Dic7 C56⋊C4 C2×C4.Dic7 C23.21D14 C22×C56 C4.Dic7 C4⋊Dic7 C23.D7 C2×C56 C22×C8 C2×C8 C2×C8 C22×C4 C14 C2×C4 C23 C2 # reps 1 2 2 1 1 1 8 4 4 8 3 12 6 3 8 18 6 48

Matrix representation of C28.12C42 in GL3(𝔽113) generated by

 112 0 0 0 8 0 0 0 99
,
 15 0 0 0 0 1 0 1 0
,
 98 0 0 0 44 0 0 0 44
`G:=sub<GL(3,GF(113))| [112,0,0,0,8,0,0,0,99],[15,0,0,0,0,1,0,1,0],[98,0,0,0,44,0,0,0,44] >;`

C28.12C42 in GAP, Magma, Sage, TeX

`C_{28}._{12}C_4^2`
`% in TeX`

`G:=Group("C28.12C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,635);`
`// by ID`

`G=gap.SmallGroup(448,635);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,100,102,18822]);`
`// Polycyclic`

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

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