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## G = D4⋊5Dic14order 448 = 26·7

### 1st semidirect product of D4 and Dic14 acting through Inn(D4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D4⋊5Dic14
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — D4×Dic7 — D4⋊5Dic14
 Lower central C7 — C2×C14 — D4⋊5Dic14
 Upper central C1 — C22 — C4×D4

Generators and relations for D45Dic14
G = < a,b,c,d | a4=b2=c28=1, d2=c14, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >

Subgroups: 884 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, D43Q8, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, D4×C14, C4×Dic14, C28.6Q8, C22⋊Dic14, C28⋊Q8, C28.3Q8, C2×Dic7⋊C4, C28.48D4, D4×Dic7, D4×C28, D45Dic14
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C22×Q8, C2×C4○D4, 2+ 1+4, Dic14, C22×D7, D43Q8, C2×Dic14, C23×D7, C22×Dic14, D46D14, D7×C4○D4, D45Dic14

Smallest permutation representation of D45Dic14
On 224 points
Generators in S224
(1 222 189 64)(2 65 190 223)(3 224 191 66)(4 67 192 197)(5 198 193 68)(6 69 194 199)(7 200 195 70)(8 71 196 201)(9 202 169 72)(10 73 170 203)(11 204 171 74)(12 75 172 205)(13 206 173 76)(14 77 174 207)(15 208 175 78)(16 79 176 209)(17 210 177 80)(18 81 178 211)(19 212 179 82)(20 83 180 213)(21 214 181 84)(22 57 182 215)(23 216 183 58)(24 59 184 217)(25 218 185 60)(26 61 186 219)(27 220 187 62)(28 63 188 221)(29 134 155 109)(30 110 156 135)(31 136 157 111)(32 112 158 137)(33 138 159 85)(34 86 160 139)(35 140 161 87)(36 88 162 113)(37 114 163 89)(38 90 164 115)(39 116 165 91)(40 92 166 117)(41 118 167 93)(42 94 168 119)(43 120 141 95)(44 96 142 121)(45 122 143 97)(46 98 144 123)(47 124 145 99)(48 100 146 125)(49 126 147 101)(50 102 148 127)(51 128 149 103)(52 104 150 129)(53 130 151 105)(54 106 152 131)(55 132 153 107)(56 108 154 133)
(2 190)(4 192)(6 194)(8 196)(10 170)(12 172)(14 174)(16 176)(18 178)(20 180)(22 182)(24 184)(26 186)(28 188)(30 156)(32 158)(34 160)(36 162)(38 164)(40 166)(42 168)(44 142)(46 144)(48 146)(50 148)(52 150)(54 152)(56 154)(58 216)(60 218)(62 220)(64 222)(66 224)(68 198)(70 200)(72 202)(74 204)(76 206)(78 208)(80 210)(82 212)(84 214)(85 138)(87 140)(89 114)(91 116)(93 118)(95 120)(97 122)(99 124)(101 126)(103 128)(105 130)(107 132)(109 134)(111 136)
(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 36 15 50)(2 35 16 49)(3 34 17 48)(4 33 18 47)(5 32 19 46)(6 31 20 45)(7 30 21 44)(8 29 22 43)(9 56 23 42)(10 55 24 41)(11 54 25 40)(12 53 26 39)(13 52 27 38)(14 51 28 37)(57 120 71 134)(58 119 72 133)(59 118 73 132)(60 117 74 131)(61 116 75 130)(62 115 76 129)(63 114 77 128)(64 113 78 127)(65 140 79 126)(66 139 80 125)(67 138 81 124)(68 137 82 123)(69 136 83 122)(70 135 84 121)(85 211 99 197)(86 210 100 224)(87 209 101 223)(88 208 102 222)(89 207 103 221)(90 206 104 220)(91 205 105 219)(92 204 106 218)(93 203 107 217)(94 202 108 216)(95 201 109 215)(96 200 110 214)(97 199 111 213)(98 198 112 212)(141 196 155 182)(142 195 156 181)(143 194 157 180)(144 193 158 179)(145 192 159 178)(146 191 160 177)(147 190 161 176)(148 189 162 175)(149 188 163 174)(150 187 164 173)(151 186 165 172)(152 185 166 171)(153 184 167 170)(154 183 168 169)

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L ··· 4Q 7A 7B 7C 14A ··· 14I 14J ··· 14U 28A ··· 28L 28M ··· 28AJ order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 14 14 14 14 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D7 C4○D4 D14 D14 D14 D14 D14 Dic14 2+ 1+4 D4⋊6D14 D7×C4○D4 kernel D4⋊5Dic14 C4×Dic14 C28.6Q8 C22⋊Dic14 C28⋊Q8 C28.3Q8 C2×Dic7⋊C4 C28.48D4 D4×Dic7 D4×C28 C7×D4 C4×D4 Dic7 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C14 C2 C2 # reps 1 1 1 4 1 1 2 2 2 1 4 3 4 3 6 3 6 3 24 1 6 6

Matrix representation of D45Dic14 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 1 6 0 0 19 28
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 19 28
,
 23 21 0 0 8 25 0 0 0 0 17 15 0 0 0 12
,
 12 3 0 0 0 17 0 0 0 0 28 23 0 0 10 1
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,19,0,0,6,28],[1,0,0,0,0,1,0,0,0,0,1,19,0,0,0,28],[23,8,0,0,21,25,0,0,0,0,17,0,0,0,15,12],[12,0,0,0,3,17,0,0,0,0,28,10,0,0,23,1] >;

D45Dic14 in GAP, Magma, Sage, TeX

D_4\rtimes_5{\rm Dic}_{14}
% in TeX

G:=Group("D4:5Dic14");
// GroupNames label

G:=SmallGroup(448,992);
// by ID

G=gap.SmallGroup(448,992);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,387,675,192,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^28=1,d^2=c^14,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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