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## G = C22⋊Dic28order 448 = 26·7

### The semidirect product of C22 and Dic28 acting via Dic28/Dic14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C22⋊Dic28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C22×Dic14 — C22⋊Dic28
 Lower central C7 — C14 — C2×C28 — C22⋊Dic28
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for C22⋊Dic28
G = < a,b,c,d | a2=b2=c56=1, d2=c28, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 764 in 148 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C56, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C22⋊Q16, Dic28, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C2×Dic14, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C28.44D4, C7×C22⋊C8, C2×Dic28, C28.48D4, C22×Dic14, C22⋊Dic28
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C22≀C2, C2×Q16, C8.C22, D28, C22×D7, C22⋊Q16, Dic28, C2×D28, D4×D7, C22⋊D28, C2×Dic28, C8.D14, C22⋊Dic28

Smallest permutation representation of C22⋊Dic28
On 224 points
Generators in S224
(2 200)(4 202)(6 204)(8 206)(10 208)(12 210)(14 212)(16 214)(18 216)(20 218)(22 220)(24 222)(26 224)(28 170)(30 172)(32 174)(34 176)(36 178)(38 180)(40 182)(42 184)(44 186)(46 188)(48 190)(50 192)(52 194)(54 196)(56 198)(58 134)(60 136)(62 138)(64 140)(66 142)(68 144)(70 146)(72 148)(74 150)(76 152)(78 154)(80 156)(82 158)(84 160)(86 162)(88 164)(90 166)(92 168)(94 114)(96 116)(98 118)(100 120)(102 122)(104 124)(106 126)(108 128)(110 130)(112 132)
(1 199)(2 200)(3 201)(4 202)(5 203)(6 204)(7 205)(8 206)(9 207)(10 208)(11 209)(12 210)(13 211)(14 212)(15 213)(16 214)(17 215)(18 216)(19 217)(20 218)(21 219)(22 220)(23 221)(24 222)(25 223)(26 224)(27 169)(28 170)(29 171)(30 172)(31 173)(32 174)(33 175)(34 176)(35 177)(36 178)(37 179)(38 180)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 191)(50 192)(51 193)(52 194)(53 195)(54 196)(55 197)(56 198)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(81 157)(82 158)(83 159)(84 160)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(91 167)(92 168)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)
(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)
(1 105 29 77)(2 104 30 76)(3 103 31 75)(4 102 32 74)(5 101 33 73)(6 100 34 72)(7 99 35 71)(8 98 36 70)(9 97 37 69)(10 96 38 68)(11 95 39 67)(12 94 40 66)(13 93 41 65)(14 92 42 64)(15 91 43 63)(16 90 44 62)(17 89 45 61)(18 88 46 60)(19 87 47 59)(20 86 48 58)(21 85 49 57)(22 84 50 112)(23 83 51 111)(24 82 52 110)(25 81 53 109)(26 80 54 108)(27 79 55 107)(28 78 56 106)(113 183 141 211)(114 182 142 210)(115 181 143 209)(116 180 144 208)(117 179 145 207)(118 178 146 206)(119 177 147 205)(120 176 148 204)(121 175 149 203)(122 174 150 202)(123 173 151 201)(124 172 152 200)(125 171 153 199)(126 170 154 198)(127 169 155 197)(128 224 156 196)(129 223 157 195)(130 222 158 194)(131 221 159 193)(132 220 160 192)(133 219 161 191)(134 218 162 190)(135 217 163 189)(136 216 164 188)(137 215 165 187)(138 214 166 186)(139 213 167 185)(140 212 168 184)

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

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

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

79 conjugacy classes

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

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 Q16 D14 D14 D28 D28 Dic28 C8.C22 D4×D7 C8.D14 kernel C22⋊Dic28 C28.44D4 C7×C22⋊C8 C2×Dic28 C28.48D4 C22×Dic14 Dic14 C2×C28 C22×C14 C22⋊C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 C14 C4 C2 # reps 1 2 1 2 1 1 4 1 1 3 4 6 3 6 6 24 1 6 6

Matrix representation of C22⋊Dic28 in GL4(𝔽113) generated by

 1 0 0 0 32 112 0 0 0 0 1 0 0 0 0 1
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 112 106 0 0 81 1 0 0 0 0 70 29 0 0 84 101
,
 1 0 0 0 32 112 0 0 0 0 17 80 0 0 67 96
G:=sub<GL(4,GF(113))| [1,32,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,81,0,0,106,1,0,0,0,0,70,84,0,0,29,101],[1,32,0,0,0,112,0,0,0,0,17,67,0,0,80,96] >;

C22⋊Dic28 in GAP, Magma, Sage, TeX

C_2^2\rtimes {\rm Dic}_{28}
% in TeX

G:=Group("C2^2:Dic28");
// GroupNames label

G:=SmallGroup(448,273);
// by ID

G=gap.SmallGroup(448,273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,219,226,1123,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^56=1,d^2=c^28,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽