Dic14:7D4 - GroupNames

G = Dic₁₄⋊₇D₄ order 448 = 2⁶·7

7^th semidirect product of Dic₁₄ and D₄ acting via D₄/C₂=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₁₄⋊₇D₄, D₁₄⋊₈SD₁₆, (C₇×Q₈)⋊₅D₄, C₄.₆₃(D₄×D₇), D₁₄⋊C₈⋊₃₃C₂, Q₈⋊₂(C₇⋊D₄), C₇⋊₄(Q₈⋊D₄), C₂₈.₄₈(C₂×D₄), (C₂×SD₁₆)⋊₁₁D₇, (C₂×D₄).₇₂D₁₄, C₂₈⋊₂D₄.₇C₂, (C₂×C₈).₁₄₇D₁₄, C₂.₂₉(D₇×SD₁₆), C₁₄.₅₈C₂²≀C₂, (C₁₄×SD₁₆)⋊₂₁C₂, Q₈⋊Dic₇⋊₂₈C₂, (C₂×Q₈).₁₁₆D₁₄, (C₂×Dic₇).₇₂D₄, C₁₄.₄₆(C₂×SD₁₆), (C₂²×D₇).₉₂D₄, C₂².₂₆₇(D₄×D₇), C₂₈.₄₄D₄⋊₃₅C₂, (C₂×C₂₈).₄₄₇C₂³, (C₂×C₅₆).₂₉₄C₂², (D₄×C₁₄).₉₆C₂², (Q₈×C₁₄).₇₆C₂², C₂.₂₆(C₂³⋊D₁₄), C₂.₂₉(SD₁₆⋊D₇), C₁₄.₄₉(C₈.C₂²), C₄⋊Dic₇.₁₇₄C₂², (C₂×Dic₁₄).₁₂₇C₂², (C₂×Q₈×D₇)⋊₂C₂, C₄.₄₃(C₂×C₇⋊D₄), (C₂×D₄.D₇)⋊₂₀C₂, (C₂×C₄×D₇).₄₈C₂², (C₂×C₁₄).₃₅₉(C₂×D₄), (C₂×C₇⋊C₈).₁₅₇C₂², (C₂×C₄).₅₃₆(C₂²×D₇), SmallGroup(448,704)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — Dic₁₄⋊₇D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×C₄×D₇ — C₂×Q₈×D₇ — Dic₁₄⋊₇D₄

Lower central

C₇ — C₁₄ — C₂×C₂₈ — Dic₁₄⋊₇D₄

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for Dic₁₄⋊₇D₄
G = < a,b,c,d | a¹⁴=b²=c⁸=d²=1, bab=a^-1, ac=ca, ad=da, cbc^-1=dbd=a⁷b, dcd=c³ >

Subgroups: 836 in 158 conjugacy classes, 45 normal (37 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, Q₈⋊C₄, C₄⋊D₄, C₂×SD₁₆, C₂×SD₁₆, C₂²×Q₈, C₇⋊C₈, C₅₆, Dic₁₄, Dic₁₄, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, Q₈⋊D₄, C₂×C₇⋊C₈, C₄⋊Dic₇, D₄.D₇, C₂³.D₇, C₂×C₅₆, C₇×SD₁₆, C₂×Dic₁₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, Q₈×D₇, C₂×C₇⋊D₄, D₄×C₁₄, Q₈×C₁₄, C₂₈.₄₄D₄, D₁₄⋊C₈, Q₈⋊Dic₇, C₂×D₄.D₇, C₂₈⋊₂D₄, C₁₄×SD₁₆, C₂×Q₈×D₇, Dic₁₄⋊₇D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, SD₁₆, C₂×D₄, D₁₄, C₂²≀C₂, C₂×SD₁₆, C₈.C₂², C₇⋊D₄, C₂²×D₇, Q₈⋊D₄, D₄×D₇, C₂×C₇⋊D₄, D₇×SD₁₆, SD₁₆⋊D₇, C₂³⋊D₁₄, Dic₁₄⋊₇D₄

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

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

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14O	28A	···	28F	28G	···	28L	56A	···	56L
order	1	2	2	2	2	2	2	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	8	14	14	2	2	4	4	28	28	28	56	2	2	2	4	4	28	28	2	···	2	8	···	8	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	2	2	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+		-	+	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₄	D₇	SD₁₆	D₁₄	D₁₄	D₁₄	C₇⋊D₄	C₈.C₂²	D₄×D₇	D₄×D₇	D₇×SD₁₆	SD₁₆⋊D₇
kernel	Dic₁₄⋊₇D₄	C₂₈.₄₄D₄	D₁₄⋊C₈	Q₈⋊Dic₇	C₂×D₄.D₇	C₂₈⋊₂D₄	C₁₄×SD₁₆	C₂×Q₈×D₇	Dic₁₄	C₂×Dic₇	C₇×Q₈	C₂²×D₇	C₂×SD₁₆	D₁₄	C₂×C₈	C₂×D₄	C₂×Q₈	Q₈	C₁₄	C₄	C₂²	C₂	C₂
# reps	1	1	1	1	1	1	1	1	2	1	2	1	3	4	3	3	3	12	1	3	3	6	6

Matrix representation of Dic₁₄⋊₇D₄ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	90	103
0	0	0	0	20	102

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	1	0	0
0	0	0	0	10	112
0	0	0	0	99	103

100	13	0	0	0	0
100	100	0	0	0	0
0	0	0	1	0	0
0	0	112	0	0	0
0	0	0	0	112	0
0	0	0	0	0	112

109	18	0	0	0	0
18	4	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,90,20,0,0,0,0,103,102],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,10,99,0,0,0,0,112,103],[100,100,0,0,0,0,13,100,0,0,0,0,0,0,0,112,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[109,18,0,0,0,0,18,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic₁₄⋊₇D₄ in GAP, Magma, Sage, TeX

{\rm Dic}_{14}\rtimes_7D_4

% in TeX

G:=Group("Dic14:7D4");

// GroupNames label

G:=SmallGroup(448,704);

// by ID

G=gap.SmallGroup(448,704);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,219,851,438,102,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^7*b,d*c*d=c^3>;

// generators/relations

G = Dic14⋊7D4 order 448 = 26·7

7th semidirect product of Dic14 and D4 acting via D4/C2=C22

G = Dic₁₄⋊₇D₄ order 448 = 2⁶·7

7^th semidirect product of Dic₁₄ and D₄ acting via D₄/C₂=C₂²