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## G = D56.1C4order 448 = 26·7

### 1st non-split extension by D56 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — D56.1C4
 Chief series C1 — C7 — C14 — C28 — C56 — C2×C56 — D56⋊7C2 — D56.1C4
 Lower central C7 — C14 — C28 — C56 — D56.1C4
 Upper central C1 — C4 — C2×C4 — C2×C8 — C2×C16

Generators and relations for D56.1C4
G = < a,b,c | a56=b2=1, c4=a42, bab=a-1, ac=ca, cbc-1=a7b >

Smallest permutation representation of D56.1C4
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)
(1 118)(2 117)(3 116)(4 115)(5 114)(6 113)(7 168)(8 167)(9 166)(10 165)(11 164)(12 163)(13 162)(14 161)(15 160)(16 159)(17 158)(18 157)(19 156)(20 155)(21 154)(22 153)(23 152)(24 151)(25 150)(26 149)(27 148)(28 147)(29 146)(30 145)(31 144)(32 143)(33 142)(34 141)(35 140)(36 139)(37 138)(38 137)(39 136)(40 135)(41 134)(42 133)(43 132)(44 131)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 120)(56 119)(57 176)(58 175)(59 174)(60 173)(61 172)(62 171)(63 170)(64 169)(65 224)(66 223)(67 222)(68 221)(69 220)(70 219)(71 218)(72 217)(73 216)(74 215)(75 214)(76 213)(77 212)(78 211)(79 210)(80 209)(81 208)(82 207)(83 206)(84 205)(85 204)(86 203)(87 202)(88 201)(89 200)(90 199)(91 198)(92 197)(93 196)(94 195)(95 194)(96 193)(97 192)(98 191)(99 190)(100 189)(101 188)(102 187)(103 186)(104 185)(105 184)(106 183)(107 182)(108 181)(109 180)(110 179)(111 178)(112 177)
(1 99 50 92 43 85 36 78 29 71 22 64 15 57 8 106)(2 100 51 93 44 86 37 79 30 72 23 65 16 58 9 107)(3 101 52 94 45 87 38 80 31 73 24 66 17 59 10 108)(4 102 53 95 46 88 39 81 32 74 25 67 18 60 11 109)(5 103 54 96 47 89 40 82 33 75 26 68 19 61 12 110)(6 104 55 97 48 90 41 83 34 76 27 69 20 62 13 111)(7 105 56 98 49 91 42 84 35 77 28 70 21 63 14 112)(113 192 134 213 155 178 120 199 141 220 162 185 127 206 148 171)(114 193 135 214 156 179 121 200 142 221 163 186 128 207 149 172)(115 194 136 215 157 180 122 201 143 222 164 187 129 208 150 173)(116 195 137 216 158 181 123 202 144 223 165 188 130 209 151 174)(117 196 138 217 159 182 124 203 145 224 166 189 131 210 152 175)(118 197 139 218 160 183 125 204 146 169 167 190 132 211 153 176)(119 198 140 219 161 184 126 205 147 170 168 191 133 212 154 177)```

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

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

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

118 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 8E 8F 14A ··· 14I 16A ··· 16H 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 4 4 4 4 7 7 7 8 8 8 8 8 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 2 56 1 1 2 56 2 2 2 2 2 2 2 56 56 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 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D7 D8 SD16 D14 C4×D7 C7⋊D4 D28 D8.C4 D56 C56⋊C2 D56.1C4 kernel D56.1C4 C56.C4 C2×C112 D56⋊7C2 D56 Dic28 C56 C2×C28 C2×C16 C28 C2×C14 C2×C8 C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 3 2 2 3 6 6 6 8 12 12 48

Matrix representation of D56.1C4 in GL2(𝔽113) generated by

 22 0 0 36
,
 0 36 22 0
,
 78 0 0 65
`G:=sub<GL(2,GF(113))| [22,0,0,36],[0,22,36,0],[78,0,0,65] >;`

D56.1C4 in GAP, Magma, Sage, TeX

`D_{56}._1C_4`
`% in TeX`

`G:=Group("D56.1C4");`
`// GroupNames label`

`G:=SmallGroup(448,67);`
`// by ID`

`G=gap.SmallGroup(448,67);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,268,1123,1684,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^56=b^2=1,c^4=a^42,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;`
`// generators/relations`

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