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## G = C28.(C4○D4)  order 448 = 26·7

### 32nd non-split extension by C28 of C4○D4 acting via C4○D4/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28.(C4○D4)
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×C4×D7 — C4⋊C4⋊7D7 — C28.(C4○D4)
 Lower central C7 — C14 — C2×C28 — C28.(C4○D4)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C28.(C4○D4)
G = < a,b,c,d | a4=b4=1, c14=a2b2, d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a-1b, dcd-1=b2c13 >

Subgroups: 492 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C7⋊C8, C56, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C23.20D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×Dic14, C2×C4×D7, C28.Q8, C14.Q16, C28.44D4, D14⋊C8, C7×C4.Q8, C4⋊C47D7, D142Q8, C28.(C4○D4)
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8.C22, C22×D7, C23.20D4, C4○D28, D4×D7, Q82D7, D14.5D4, SD16⋊D7, SD163D7, C28.(C4○D4)

Smallest permutation representation of C28.(C4○D4)
On 224 points
Generators in S224
```(1 187 98 151)(2 152 99 188)(3 189 100 153)(4 154 101 190)(5 191 102 155)(6 156 103 192)(7 193 104 157)(8 158 105 194)(9 195 106 159)(10 160 107 196)(11 169 108 161)(12 162 109 170)(13 171 110 163)(14 164 111 172)(15 173 112 165)(16 166 85 174)(17 175 86 167)(18 168 87 176)(19 177 88 141)(20 142 89 178)(21 179 90 143)(22 144 91 180)(23 181 92 145)(24 146 93 182)(25 183 94 147)(26 148 95 184)(27 185 96 149)(28 150 97 186)(29 117 224 83)(30 84 197 118)(31 119 198 57)(32 58 199 120)(33 121 200 59)(34 60 201 122)(35 123 202 61)(36 62 203 124)(37 125 204 63)(38 64 205 126)(39 127 206 65)(40 66 207 128)(41 129 208 67)(42 68 209 130)(43 131 210 69)(44 70 211 132)(45 133 212 71)(46 72 213 134)(47 135 214 73)(48 74 215 136)(49 137 216 75)(50 76 217 138)(51 139 218 77)(52 78 219 140)(53 113 220 79)(54 80 221 114)(55 115 222 81)(56 82 223 116)
(1 73 112 121)(2 215 85 34)(3 75 86 123)(4 217 87 36)(5 77 88 125)(6 219 89 38)(7 79 90 127)(8 221 91 40)(9 81 92 129)(10 223 93 42)(11 83 94 131)(12 197 95 44)(13 57 96 133)(14 199 97 46)(15 59 98 135)(16 201 99 48)(17 61 100 137)(18 203 101 50)(19 63 102 139)(20 205 103 52)(21 65 104 113)(22 207 105 54)(23 67 106 115)(24 209 107 56)(25 69 108 117)(26 211 109 30)(27 71 110 119)(28 213 111 32)(29 183 210 161)(31 185 212 163)(33 187 214 165)(35 189 216 167)(37 191 218 141)(39 193 220 143)(41 195 222 145)(43 169 224 147)(45 171 198 149)(47 173 200 151)(49 175 202 153)(51 177 204 155)(53 179 206 157)(55 181 208 159)(58 186 134 164)(60 188 136 166)(62 190 138 168)(64 192 140 142)(66 194 114 144)(68 196 116 146)(70 170 118 148)(72 172 120 150)(74 174 122 152)(76 176 124 154)(78 178 126 156)(80 180 128 158)(82 182 130 160)(84 184 132 162)
(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 97 98 28)(2 27 99 96)(3 95 100 26)(4 25 101 94)(5 93 102 24)(6 23 103 92)(7 91 104 22)(8 21 105 90)(9 89 106 20)(10 19 107 88)(11 87 108 18)(12 17 109 86)(13 85 110 16)(14 15 111 112)(29 36 224 203)(30 202 197 35)(31 34 198 201)(32 200 199 33)(37 56 204 223)(38 222 205 55)(39 54 206 221)(40 220 207 53)(41 52 208 219)(42 218 209 51)(43 50 210 217)(44 216 211 49)(45 48 212 215)(46 214 213 47)(57 60 119 122)(58 121 120 59)(61 84 123 118)(62 117 124 83)(63 82 125 116)(64 115 126 81)(65 80 127 114)(66 113 128 79)(67 78 129 140)(68 139 130 77)(69 76 131 138)(70 137 132 75)(71 74 133 136)(72 135 134 73)(141 196 177 160)(142 159 178 195)(143 194 179 158)(144 157 180 193)(145 192 181 156)(146 155 182 191)(147 190 183 154)(148 153 184 189)(149 188 185 152)(150 151 186 187)(161 176 169 168)(162 167 170 175)(163 174 171 166)(164 165 172 173)```

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 C4○D8 C4○D28 C8.C22 Q8⋊2D7 D4×D7 SD16⋊D7 SD16⋊3D7 kernel C28.(C4○D4) C28.Q8 C14.Q16 C28.44D4 D14⋊C8 C7×C4.Q8 C4⋊C4⋊7D7 D14⋊2Q8 C2×Dic7 C22×D7 C4.Q8 C28 C4⋊C4 C2×C8 C14 C4 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 3 4 6 3 4 12 1 3 3 6 6

Matrix representation of C28.(C4○D4) in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 98 0 0 0 111 15
,
 15 0 0 0 0 15 0 0 0 0 95 44 0 0 26 18
,
 36 90 0 0 23 23 0 0 0 0 98 112 0 0 111 15
,
 105 17 0 0 96 8 0 0 0 0 98 112 0 0 0 15
`G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,98,111,0,0,0,15],[15,0,0,0,0,15,0,0,0,0,95,26,0,0,44,18],[36,23,0,0,90,23,0,0,0,0,98,111,0,0,112,15],[105,96,0,0,17,8,0,0,0,0,98,0,0,0,112,15] >;`

C28.(C4○D4) in GAP, Magma, Sage, TeX

`C_{28}.(C_4\circ D_4)`
`% in TeX`

`G:=Group("C28.(C4oD4)");`
`// GroupNames label`

`G:=SmallGroup(448,401);`
`// by ID`

`G=gap.SmallGroup(448,401);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,926,219,100,851,102,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^14=a^2*b^2,d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^-1*b,d*c*d^-1=b^2*c^13>;`
`// generators/relations`

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