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## G = C28⋊4D8order 448 = 26·7

### 1st semidirect product of C28 and D8 acting via D8/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28⋊4D8
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C28⋊4D4 — C28⋊4D8
 Lower central C7 — C14 — C2×C28 — C28⋊4D8
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C284D8
G = < a,b,c | a28=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1412 in 162 conjugacy classes, 55 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C42, C2×C8, D8, C2×D4, C28, D14, C2×C14, C4×C8, C41D4, C2×D8, C56, D28, C2×C28, C2×C28, C22×D7, C84D4, D56, C4×C28, C2×C56, C2×D28, C2×D28, C4×C56, C284D4, C2×D56, C284D8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C41D4, C2×D8, D28, C22×D7, C84D4, D56, C2×D28, C284D4, C2×D56, C284D8

Smallest permutation representation of C284D8
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)
(1 217 124 85 159 186 41 77)(2 218 125 86 160 187 42 78)(3 219 126 87 161 188 43 79)(4 220 127 88 162 189 44 80)(5 221 128 89 163 190 45 81)(6 222 129 90 164 191 46 82)(7 223 130 91 165 192 47 83)(8 224 131 92 166 193 48 84)(9 197 132 93 167 194 49 57)(10 198 133 94 168 195 50 58)(11 199 134 95 141 196 51 59)(12 200 135 96 142 169 52 60)(13 201 136 97 143 170 53 61)(14 202 137 98 144 171 54 62)(15 203 138 99 145 172 55 63)(16 204 139 100 146 173 56 64)(17 205 140 101 147 174 29 65)(18 206 113 102 148 175 30 66)(19 207 114 103 149 176 31 67)(20 208 115 104 150 177 32 68)(21 209 116 105 151 178 33 69)(22 210 117 106 152 179 34 70)(23 211 118 107 153 180 35 71)(24 212 119 108 154 181 36 72)(25 213 120 109 155 182 37 73)(26 214 121 110 156 183 38 74)(27 215 122 111 157 184 39 75)(28 216 123 112 158 185 40 76)
(1 8)(2 7)(3 6)(4 5)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)(29 115)(30 114)(31 113)(32 140)(33 139)(34 138)(35 137)(36 136)(37 135)(38 134)(39 133)(40 132)(41 131)(42 130)(43 129)(44 128)(45 127)(46 126)(47 125)(48 124)(49 123)(50 122)(51 121)(52 120)(53 119)(54 118)(55 117)(56 116)(57 216)(58 215)(59 214)(60 213)(61 212)(62 211)(63 210)(64 209)(65 208)(66 207)(67 206)(68 205)(69 204)(70 203)(71 202)(72 201)(73 200)(74 199)(75 198)(76 197)(77 224)(78 223)(79 222)(80 221)(81 220)(82 219)(83 218)(84 217)(85 193)(86 192)(87 191)(88 190)(89 189)(90 188)(91 187)(92 186)(93 185)(94 184)(95 183)(96 182)(97 181)(98 180)(99 179)(100 178)(101 177)(102 176)(103 175)(104 174)(105 173)(106 172)(107 171)(108 170)(109 169)(110 196)(111 195)(112 194)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)```

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

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

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

118 conjugacy classes

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

118 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D7 D8 D14 D14 D28 D28 D56 kernel C28⋊4D8 C4×C56 C28⋊4D4 C2×D56 C56 C2×C28 C4×C8 C28 C42 C2×C8 C8 C2×C4 C4 # reps 1 1 2 4 4 2 3 8 3 6 24 12 48

Matrix representation of C284D8 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 90 81 0 0 13 77
,
 82 31 0 0 82 82 0 0 0 0 69 65 0 0 76 106
,
 1 0 0 0 0 112 0 0 0 0 23 58 0 0 100 90
`G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,90,13,0,0,81,77],[82,82,0,0,31,82,0,0,0,0,69,76,0,0,65,106],[1,0,0,0,0,112,0,0,0,0,23,100,0,0,58,90] >;`

C284D8 in GAP, Magma, Sage, TeX

`C_{28}\rtimes_4D_8`
`% in TeX`

`G:=Group("C28:4D8");`
`// GroupNames label`

`G:=SmallGroup(448,229);`
`// by ID`

`G=gap.SmallGroup(448,229);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,226,1123,136,18822]);`
`// Polycyclic`

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

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