Dic7:M4(2) - GroupNames

G = Dic₇⋊M₄(2) order 448 = 2⁶·7

1^st semidirect product of Dic₇ and M₄(2) acting via M₄(2)/C₈=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic₇⋊₁M₄(2), C₇⋊C₈⋊₁₆D₄, C₇⋊₂(C₈⋊₆D₄), D₁₄⋊C₈⋊₂₀C₂, D₁₄⋊C₄.₆C₄, C₂²⋊C₈⋊₁₄D₇, C₁₄.₂₈(C₄×D₄), C₄.₁₉₈(D₄×D₇), Dic₇⋊C₈⋊₂₀C₂, (C₈×Dic₇)⋊₁₆C₂, C₂₈.₃₅₇(C₂×D₄), (C₂×C₈).₁₉₆D₁₄, Dic₇⋊C₄.₆C₄, C₂³.₁₄(C₄×D₇), C₂³.D₇.₆C₄, C₁₄.₁₀(C₈○D₄), (C₂²×C₄).₈₀D₁₄, C₂.₁₄(D₇×M₄(2)), C₂₈.₂₉₉(C₄○D₄), (C₂×C₂₈).₈₂₄C₂³, (C₂×C₅₆).₁₇₃C₂², C₁₄.₂₃(C₂×M₄(2)), C₄.₁₂₅(D₄⋊₂D₇), (C₂²×C₂₈).₉₅C₂², C₂.₁₂(Dic₇⋊₄D₄), C₂.₁₂(D₂₈.₂C₄), (C₄×Dic₇).₂₇₂C₂², (C₂×C₄).₃₃(C₄×D₇), (C₄×C₇⋊D₄).₂C₂, (C₂×C₇⋊D₄).₆C₄, (C₂×C₈⋊D₇)⋊₁₃C₂, (C₇×C₂²⋊C₈)⋊₁₈C₂, (C₂×C₂₈).₄₁(C₂×C₄), (C₂×C₄.Dic₇)⋊₂C₂, C₂².₁₀₆(C₂×C₄×D₇), (C₂×C₇⋊C₈).₁₉₂C₂², (C₂×C₄×D₇).₁₇₉C₂², (C₂²×C₁₄).₄₂(C₂×C₄), (C₂×C₁₄).₇₉(C₂²×C₄), (C₂×Dic₇).₅₁(C₂×C₄), (C₂²×D₇).₁₃(C₂×C₄), (C₂×C₄).₇₆₆(C₂²×D₇), SmallGroup(448,263)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — Dic₇⋊M₄(2)

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₂×C₄×D₇ — C₄×C₇⋊D₄ — Dic₇⋊M₄(2)

Lower central

C₇ — C₂×C₁₄ — Dic₇⋊M₄(2)

Upper central

C₁ — C₂×C₄ — C₂²⋊C₈

Generators and relations for Dic₇⋊M₄(2)
G = < a,b,c,d | a¹⁴=c⁸=d²=1, b²=a⁷, bab^-1=cac^-1=a^-1, ad=da, cbc^-1=dbd=a⁷b, dcd=c⁵ >

Subgroups: 508 in 122 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, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄×C₈, C₂²⋊C₈, C₂²⋊C₈, C₄⋊C₈, C₄×D₄, C₂×M₄(2), C₇⋊C₈, C₇⋊C₈, C₅₆, C₄×D₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂²×C₁₄, C₈⋊₆D₄, 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, C₈×Dic₇, Dic₇⋊C₈, D₁₄⋊C₈, C₇×C₂²⋊C₈, C₂×C₈⋊D₇, C₂×C₄.Dic₇, C₄×C₇⋊D₄, Dic₇⋊M₄(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⋊₆D₄, C₂×C₄×D₇, D₄×D₇, D₄⋊₂D₇, Dic₇⋊₄D₄, D₂₈.₂C₄, D₇×M₄(2), Dic₇⋊M₄(2)

Smallest permutation representation of Dic₇⋊M₄(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 69 8 62)(2 68 9 61)(3 67 10 60)(4 66 11 59)(5 65 12 58)(6 64 13 57)(7 63 14 70)(15 111 22 104)(16 110 23 103)(17 109 24 102)(18 108 25 101)(19 107 26 100)(20 106 27 99)(21 105 28 112)(29 151 36 144)(30 150 37 143)(31 149 38 142)(32 148 39 141)(33 147 40 154)(34 146 41 153)(35 145 42 152)(43 89 50 96)(44 88 51 95)(45 87 52 94)(46 86 53 93)(47 85 54 92)(48 98 55 91)(49 97 56 90)(71 200 78 207)(72 199 79 206)(73 198 80 205)(74 197 81 204)(75 210 82 203)(76 209 83 202)(77 208 84 201)(113 193 120 186)(114 192 121 185)(115 191 122 184)(116 190 123 183)(117 189 124 196)(118 188 125 195)(119 187 126 194)(127 212 134 219)(128 211 135 218)(129 224 136 217)(130 223 137 216)(131 222 138 215)(132 221 139 214)(133 220 140 213)(155 170 162 177)(156 169 163 176)(157 182 164 175)(158 181 165 174)(159 180 166 173)(160 179 167 172)(161 178 168 171)

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

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

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

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

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

88 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	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	4	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	28	1	1	1	1	4	14	14	14	14	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	M₄(2)	C₄○D₄	D₁₄	D₁₄	C₈○D₄	C₄×D₇	C₄×D₇	D₂₈.₂C₄	D₄×D₇	D₄⋊₂D₇	D₇×M₄(2)
kernel	Dic₇⋊M₄(2)	C₈×Dic₇	Dic₇⋊C₈	D₁₄⋊C₈	C₇×C₂²⋊C₈	C₂×C₈⋊D₇	C₂×C₄.Dic₇	C₄×C₇⋊D₄	Dic₇⋊C₄	D₁₄⋊C₄	C₂³.D₇	C₂×C₇⋊D₄	C₇⋊C₈	C₂²⋊C₈	Dic₇	C₂₈	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	4	2	6	3	4	6	6	24	3	3	6

Matrix representation of Dic₇⋊M₄(2) ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	10	1	0	0
0	0	102	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

1	111	0	0	0	0
1	112	0	0	0	0
0	0	0	80	0	0
0	0	89	0	0	0
0	0	0	0	95	44
0	0	0	0	26	18

18	0	0	0	0	0
18	95	0	0	0	0
0	0	0	33	0	0
0	0	24	0	0	0
0	0	0	0	95	44
0	0	0	0	62	18

1	111	0	0	0	0
0	112	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	18	69
0	0	0	0	51	95

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,10,102,0,0,0,0,1,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,1,0,0,0,0,111,112,0,0,0,0,0,0,0,89,0,0,0,0,80,0,0,0,0,0,0,0,95,26,0,0,0,0,44,18],[18,18,0,0,0,0,0,95,0,0,0,0,0,0,0,24,0,0,0,0,33,0,0,0,0,0,0,0,95,62,0,0,0,0,44,18],[1,0,0,0,0,0,111,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,51,0,0,0,0,69,95] >;

Dic₇⋊M₄(2) in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes M_4(2)

% in TeX

G:=Group("Dic7:M4(2)");

// GroupNames label

G:=SmallGroup(448,263);

// by ID

G=gap.SmallGroup(448,263);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,701,422,219,58,136,18822]);

// Polycyclic

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

// generators/relations

G = Dic7⋊M4(2) order 448 = 26·7

1st semidirect product of Dic7 and M4(2) acting via M4(2)/C8=C2

G = Dic₇⋊M₄(2) order 448 = 2⁶·7

1^st semidirect product of Dic₇ and M₄(2) acting via M₄(2)/C₈=C₂