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## G = C8.8D28order 448 = 26·7

### 4th non-split extension by C8 of D28 acting via D28/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C8.8D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C4.D28 — C8.8D28
 Lower central C7 — C14 — C2×C28 — C8.8D28
 Upper central C1 — C22 — C42 — C4×C8

Generators and relations for C8.8D28
G = < a,b,c | a8=b28=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 900 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C7, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], D7 [×2], C14, C14 [×2], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×6], C2×C14, C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C56 [×4], Dic14 [×4], D28 [×4], C2×Dic7 [×2], C2×C28, C2×C28 [×2], C22×D7 [×2], C8.12D4, C56⋊C2 [×4], D56 [×2], Dic28 [×2], D14⋊C4 [×4], C4×C28, C2×C56 [×2], C2×Dic14 [×2], C2×D28 [×2], C4×C56, C4.D28 [×2], C2×C56⋊C2 [×2], C2×D56, C2×Dic28, C8.8D28
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D7, C2×D4 [×3], D14 [×3], C41D4, C4○D8 [×2], D28 [×6], C22×D7, C8.12D4, C2×D28 [×3], C284D4, D567C2 [×2], C8.8D28

Smallest permutation representation of C8.8D28
On 224 points
Generators in S224
```(1 166 37 211 182 86 69 132)(2 167 38 212 183 87 70 133)(3 168 39 213 184 88 71 134)(4 141 40 214 185 89 72 135)(5 142 41 215 186 90 73 136)(6 143 42 216 187 91 74 137)(7 144 43 217 188 92 75 138)(8 145 44 218 189 93 76 139)(9 146 45 219 190 94 77 140)(10 147 46 220 191 95 78 113)(11 148 47 221 192 96 79 114)(12 149 48 222 193 97 80 115)(13 150 49 223 194 98 81 116)(14 151 50 224 195 99 82 117)(15 152 51 197 196 100 83 118)(16 153 52 198 169 101 84 119)(17 154 53 199 170 102 57 120)(18 155 54 200 171 103 58 121)(19 156 55 201 172 104 59 122)(20 157 56 202 173 105 60 123)(21 158 29 203 174 106 61 124)(22 159 30 204 175 107 62 125)(23 160 31 205 176 108 63 126)(24 161 32 206 177 109 64 127)(25 162 33 207 178 110 65 128)(26 163 34 208 179 111 66 129)(27 164 35 209 180 112 67 130)(28 165 36 210 181 85 68 131)
(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)
(1 210 182 131)(2 130 183 209)(3 208 184 129)(4 128 185 207)(5 206 186 127)(6 126 187 205)(7 204 188 125)(8 124 189 203)(9 202 190 123)(10 122 191 201)(11 200 192 121)(12 120 193 199)(13 198 194 119)(14 118 195 197)(15 224 196 117)(16 116 169 223)(17 222 170 115)(18 114 171 221)(19 220 172 113)(20 140 173 219)(21 218 174 139)(22 138 175 217)(23 216 176 137)(24 136 177 215)(25 214 178 135)(26 134 179 213)(27 212 180 133)(28 132 181 211)(29 145 61 93)(30 92 62 144)(31 143 63 91)(32 90 64 142)(33 141 65 89)(34 88 66 168)(35 167 67 87)(36 86 68 166)(37 165 69 85)(38 112 70 164)(39 163 71 111)(40 110 72 162)(41 161 73 109)(42 108 74 160)(43 159 75 107)(44 106 76 158)(45 157 77 105)(46 104 78 156)(47 155 79 103)(48 102 80 154)(49 153 81 101)(50 100 82 152)(51 151 83 99)(52 98 84 150)(53 149 57 97)(54 96 58 148)(55 147 59 95)(56 94 60 146)```

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

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

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

118 conjugacy classes

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

118 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C4○D8 D28 D28 D56⋊7C2 kernel C8.8D28 C4×C56 C4.D28 C2×C56⋊C2 C2×D56 C2×Dic28 C56 C2×C28 C4×C8 C42 C2×C8 C14 C8 C2×C4 C2 # reps 1 1 2 2 1 1 4 2 3 3 6 8 24 12 48

Matrix representation of C8.8D28 in GL4(𝔽113) generated by

 21 70 0 0 93 41 0 0 0 0 21 91 0 0 2 41
,
 85 82 0 0 25 60 0 0 0 0 22 92 0 0 43 0
,
 42 78 0 0 2 71 0 0 0 0 71 43 0 0 1 42
`G:=sub<GL(4,GF(113))| [21,93,0,0,70,41,0,0,0,0,21,2,0,0,91,41],[85,25,0,0,82,60,0,0,0,0,22,43,0,0,92,0],[42,2,0,0,78,71,0,0,0,0,71,1,0,0,43,42] >;`

C8.8D28 in GAP, Magma, Sage, TeX

`C_8._8D_{28}`
`% in TeX`

`G:=Group("C8.8D28");`
`// GroupNames label`

`G:=SmallGroup(448,230);`
`// by ID`

`G=gap.SmallGroup(448,230);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,58,1123,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^28=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;`
`// generators/relations`

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