D14:2M4(2) - GroupNames

G = D₁₄⋊₂M₄(2) order 448 = 2⁶·7

2^nd semidirect product of D₁₄ and M₄(2) acting via M₄(2)/C₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₁₄⋊₂M₄(2), C₇⋊C₈⋊₂₅D₄, C₇⋊₂(C₈⋊₉D₄), D₁₄⋊C₈⋊₁₉C₂, D₁₄⋊C₄.₅C₄, C₂²⋊C₈⋊₁₃D₇, C₅₆⋊C₄⋊₁₃C₂, C₄.₁₉₇(D₄×D₇), C₁₄.₂₇(C₄×D₄), Dic₇⋊C₈⋊₁₉C₂, C₁₄.₉(C₈○D₄), C₂₈.₃₅₆(C₂×D₄), (C₂×C₈).₁₉₅D₁₄, Dic₇⋊C₄.₅C₄, C₂³.₁₃(C₄×D₇), C₂³.D₇.₅C₄, (C₂²×C₄).₇₉D₁₄, C₂.₁₃(D₇×M₄(2)), C₂₈.₂₉₈(C₄○D₄), (C₂×C₅₆).₁₇₂C₂², (C₂×C₂₈).₈₂₃C₂³, C₁₄.₂₂(C₂×M₄(2)), C₄.₁₂₄(D₄⋊₂D₇), (C₂²×C₂₈).₉₄C₂², C₂.₁₁(Dic₇⋊₄D₄), C₂.₁₁(D₂₈.₂C₄), (C₄×Dic₇).₁₈₂C₂², (D₇×C₂×C₈)⋊₁₅C₂, (C₂×C₄).₃₂(C₄×D₇), (C₄×C₇⋊D₄).₁C₂, (C₂×C₇⋊D₄).₅C₄, (C₇×C₂²⋊C₈)⋊₁₇C₂, (C₂×C₂₈).₄₀(C₂×C₄), (C₂×C₄.Dic₇)⋊₁C₂, C₂².₁₀₅(C₂×C₄×D₇), (C₂×C₇⋊C₈).₁₉₁C₂², (C₂×C₄×D₇).₂₇₅C₂², (C₂²×C₁₄).₄₁(C₂×C₄), (C₂×C₁₄).₇₈(C₂²×C₄), (C₂×Dic₇).₁₇(C₂×C₄), (C₂²×D₇).₃₆(C₂×C₄), (C₂×C₄).₇₆₅(C₂²×D₇), SmallGroup(448,262)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₁₄⋊₂M₄(2)

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×C₄×D₇ — C₄×C₇⋊D₄ — D₁₄⋊₂M₄(2)

Lower central

C₇ — C₂×C₁₄ — D₁₄⋊₂M₄(2)

Upper central

C₁ — C₂×C₄ — C₂²⋊C₈

Generators and relations for D₁₄⋊₂M₄(2)
G = < a,b,c,d | a¹⁴=b²=c⁸=d²=1, bab=cac^-1=a^-1, ad=da, cbc^-1=a⁵b, dbd=a⁷b, dcd=c⁵ >

Subgroups: 508 in 124 conjugacy classes, 51 normal (47 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₈⋊C₄, C₂²⋊C₈, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂²×C₈, C₂×M₄(2), C₇⋊C₈, C₇⋊C₈, C₅₆, C₄×D₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂²×C₁₄, C₈⋊₉D₄, C₈×D₇, C₂×C₇⋊C₈, C₄.Dic₇, C₄×Dic₇, Dic₇⋊C₄, D₁₄⋊C₄, C₂³.D₇, C₂×C₅₆, C₂×C₄×D₇, C₂×C₇⋊D₄, C₂²×C₂₈, Dic₇⋊C₈, C₅₆⋊C₄, D₁₄⋊C₈, C₇×C₂²⋊C₈, D₇×C₂×C₈, C₂×C₄.Dic₇, C₄×C₇⋊D₄, D₁₄⋊₂M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, D₇, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, D₁₄, C₄×D₄, C₂×M₄(2), C₈○D₄, C₄×D₇, C₂²×D₇, C₈⋊₉D₄, C₂×C₄×D₇, D₄×D₇, D₄⋊₂D₇, Dic₇⋊₄D₄, D₂₈.₂C₄, D₇×M₄(2), D₁₄⋊₂M₄(2)

Smallest permutation representation of D₁₄⋊₂M₄(2)
►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 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 60)(9 59)(10 58)(11 57)(12 70)(13 69)(14 68)(15 122)(16 121)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 126)(26 125)(27 124)(28 123)(29 215)(30 214)(31 213)(32 212)(33 211)(34 224)(35 223)(36 222)(37 221)(38 220)(39 219)(40 218)(41 217)(42 216)(43 197)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 200)(55 199)(56 198)(71 149)(72 148)(73 147)(74 146)(75 145)(76 144)(77 143)(78 142)(79 141)(80 154)(81 153)(82 152)(83 151)(84 150)(85 164)(86 163)(87 162)(88 161)(89 160)(90 159)(91 158)(92 157)(93 156)(94 155)(95 168)(96 167)(97 166)(98 165)(99 195)(100 194)(101 193)(102 192)(103 191)(104 190)(105 189)(106 188)(107 187)(108 186)(109 185)(110 184)(111 183)(112 196)(127 180)(128 179)(129 178)(130 177)(131 176)(132 175)(133 174)(134 173)(135 172)(136 171)(137 170)(138 169)(139 182)(140 181)

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

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

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

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

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

88 conjugacy classes

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

88 irreducible representations

dim	1	1	1	1	1	1	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	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₇	C₄○D₄	M₄(2)	D₁₄	D₁₄	C₈○D₄	C₄×D₇	C₄×D₇	D₂₈.₂C₄	D₄×D₇	D₄⋊₂D₇	D₇×M₄(2)
kernel	D₁₄⋊₂M₄(2)	Dic₇⋊C₈	C₅₆⋊C₄	D₁₄⋊C₈	C₇×C₂²⋊C₈	D₇×C₂×C₈	C₂×C₄.Dic₇	C₄×C₇⋊D₄	Dic₇⋊C₄	D₁₄⋊C₄	C₂³.D₇	C₂×C₇⋊D₄	C₇⋊C₈	C₂²⋊C₈	C₂₈	D₁₄	C₂×C₈	C₂²×C₄	C₁₄	C₂×C₄	C₂³	C₂	C₄	C₄	C₂
# reps	1	1	1	1	1	1	1	1	2	2	2	2	2	3	2	4	6	3	4	6	6	24	3	3	6

Matrix representation of D₁₄⋊₂M₄(2) ►in GL₆(𝔽₁₁₃)

10	88	0	0	0	0
14	112	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	0	0	0	1

24	79	0	0	0	0
100	89	0	0	0	0
0	0	1	0	0	0
0	0	90	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

0	55	0	0	0	0
76	0	0	0	0	0
0	0	46	4	0	0
0	0	4	67	0	0
0	0	0	0	94	61
0	0	0	0	11	19

1	0	0	0	0	0
0	1	0	0	0	0
0	0	37	72	0	0
0	0	94	76	0	0
0	0	0	0	85	7
0	0	0	0	98	28

G:=sub<GL(6,GF(113))| [10,14,0,0,0,0,88,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,100,0,0,0,0,79,89,0,0,0,0,0,0,1,90,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[0,76,0,0,0,0,55,0,0,0,0,0,0,0,46,4,0,0,0,0,4,67,0,0,0,0,0,0,94,11,0,0,0,0,61,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,94,0,0,0,0,72,76,0,0,0,0,0,0,85,98,0,0,0,0,7,28] >;

D₁₄⋊₂M₄(2) in GAP, Magma, Sage, TeX

D_{14}\rtimes_2M_4(2)

% in TeX

G:=Group("D14:2M4(2)");

// GroupNames label

G:=SmallGroup(448,262);

// by ID

G=gap.SmallGroup(448,262);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,701,219,58,136,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^14=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,d*b*d=a^7*b,d*c*d=c^5>;

// generators/relations

G = D14⋊2M4(2) order 448 = 26·7

2nd semidirect product of D14 and M4(2) acting via M4(2)/C8=C2

G = D₁₄⋊₂M₄(2) order 448 = 2⁶·7

2^nd semidirect product of D₁₄ and M₄(2) acting via M₄(2)/C₈=C₂