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## G = D4.3Dic14order 448 = 26·7

### The non-split extension by D4 of Dic14 acting via Dic14/C28=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — D4.3Dic14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C28.6Q8 — D4.3Dic14
 Lower central C7 — C14 — C2×C28 — D4.3Dic14
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for D4.3Dic14
G = < a,b,c,d | a4=b2=c28=1, d2=a2c14, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 372 in 102 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4.Q8, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C28⋊C8, C28.Q8, C4.Dic14, D4⋊Dic7, C28.6Q8, D4×C28, D4.3Dic14
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C4○D8, C8⋊C22, Dic14, C7⋊D4, C22×D7, D4.Q8, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, D4.D14, D4.8D14, D4.3Dic14

Smallest permutation representation of D4.3Dic14
On 224 points
Generators in S224
(1 65 190 134)(2 66 191 135)(3 67 192 136)(4 68 193 137)(5 69 194 138)(6 70 195 139)(7 71 196 140)(8 72 169 113)(9 73 170 114)(10 74 171 115)(11 75 172 116)(12 76 173 117)(13 77 174 118)(14 78 175 119)(15 79 176 120)(16 80 177 121)(17 81 178 122)(18 82 179 123)(19 83 180 124)(20 84 181 125)(21 57 182 126)(22 58 183 127)(23 59 184 128)(24 60 185 129)(25 61 186 130)(26 62 187 131)(27 63 188 132)(28 64 189 133)(29 162 215 108)(30 163 216 109)(31 164 217 110)(32 165 218 111)(33 166 219 112)(34 167 220 85)(35 168 221 86)(36 141 222 87)(37 142 223 88)(38 143 224 89)(39 144 197 90)(40 145 198 91)(41 146 199 92)(42 147 200 93)(43 148 201 94)(44 149 202 95)(45 150 203 96)(46 151 204 97)(47 152 205 98)(48 153 206 99)(49 154 207 100)(50 155 208 101)(51 156 209 102)(52 157 210 103)(53 158 211 104)(54 159 212 105)(55 160 213 106)(56 161 214 107)
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 215)(30 216)(31 217)(32 218)(33 219)(34 220)(35 221)(36 222)(37 223)(38 224)(39 197)(40 198)(41 199)(42 200)(43 201)(44 202)(45 203)(46 204)(47 205)(48 206)(49 207)(50 208)(51 209)(52 210)(53 211)(54 212)(55 213)(56 214)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 196)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 179)(83 180)(84 181)
(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 39 176 211)(2 224 177 52)(3 37 178 209)(4 222 179 50)(5 35 180 207)(6 220 181 48)(7 33 182 205)(8 218 183 46)(9 31 184 203)(10 216 185 44)(11 29 186 201)(12 214 187 42)(13 55 188 199)(14 212 189 40)(15 53 190 197)(16 210 191 38)(17 51 192 223)(18 208 193 36)(19 49 194 221)(20 206 195 34)(21 47 196 219)(22 204 169 32)(23 45 170 217)(24 202 171 30)(25 43 172 215)(26 200 173 56)(27 41 174 213)(28 198 175 54)(57 98 140 166)(58 151 113 111)(59 96 114 164)(60 149 115 109)(61 94 116 162)(62 147 117 107)(63 92 118 160)(64 145 119 105)(65 90 120 158)(66 143 121 103)(67 88 122 156)(68 141 123 101)(69 86 124 154)(70 167 125 99)(71 112 126 152)(72 165 127 97)(73 110 128 150)(74 163 129 95)(75 108 130 148)(76 161 131 93)(77 106 132 146)(78 159 133 91)(79 104 134 144)(80 157 135 89)(81 102 136 142)(82 155 137 87)(83 100 138 168)(84 153 139 85)

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

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

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

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 ··· 14U 28A ··· 28L 28M ··· 28AJ 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 size 1 1 1 1 4 4 2 2 2 2 4 4 4 56 56 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C2 D4 Q8 D7 C4○D4 D14 D14 D14 C4○D8 C7⋊D4 Dic14 C4○D28 C8⋊C22 D4.D14 D4.8D14 kernel D4.3Dic14 C28⋊C8 C28.Q8 C4.Dic14 D4⋊Dic7 C28.6Q8 D4×C28 C2×C28 C7×D4 C4×D4 C28 C42 C4⋊C4 C2×D4 C14 C2×C4 D4 C4 C14 C2 C2 # reps 1 1 1 1 2 1 1 2 2 3 2 3 3 3 4 12 12 12 1 6 6

Matrix representation of D4.3Dic14 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 31 91 0 0 54 82
,
 1 0 0 0 0 1 0 0 0 0 31 91 0 0 95 82
,
 21 17 0 0 79 64 0 0 0 0 15 0 0 0 0 15
,
 57 95 0 0 55 56 0 0 0 0 87 4 0 0 29 26
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,31,54,0,0,91,82],[1,0,0,0,0,1,0,0,0,0,31,95,0,0,91,82],[21,79,0,0,17,64,0,0,0,0,15,0,0,0,0,15],[57,55,0,0,95,56,0,0,0,0,87,29,0,0,4,26] >;

D4.3Dic14 in GAP, Magma, Sage, TeX

D_4._3{\rm Dic}_{14}
% in TeX

G:=Group("D4.3Dic14");
// GroupNames label

G:=SmallGroup(448,543);
// by ID

G=gap.SmallGroup(448,543);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,344,254,1123,297,136,18822]);
// Polycyclic

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

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