Copied to
clipboard

## G = M4(2)⋊Dic7order 448 = 26·7

### 2nd semidirect product of M4(2) and Dic7 acting via Dic7/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — M4(2)⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C22×C28 — C2×C4⋊Dic7 — M4(2)⋊Dic7
 Lower central C7 — C14 — C2×C14 — M4(2)⋊Dic7
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for M4(2)⋊Dic7
G = < a,b,c,d | a8=b2=c14=1, d2=c7, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 388 in 98 conjugacy classes, 51 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2×C4⋊C4, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C22×C14, C22.C42, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4⋊Dic7, C2×C56, C7×M4(2), C7×M4(2), C22×Dic7, C22×C28, C2×C4.Dic7, C2×C4⋊Dic7, C14×M4(2), M4(2)⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C4.D4, C4.10D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C22.C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C28.46D4, C4.12D28, C14.C42, M4(2)⋊Dic7

Smallest permutation representation of M4(2)⋊Dic7
On 224 points
Generators in S224
(1 111 223 195 67 201 88 175)(2 112 224 196 68 202 89 176)(3 99 211 183 69 203 90 177)(4 100 212 184 70 204 91 178)(5 101 213 185 57 205 92 179)(6 102 214 186 58 206 93 180)(7 103 215 187 59 207 94 181)(8 104 216 188 60 208 95 182)(9 105 217 189 61 209 96 169)(10 106 218 190 62 210 97 170)(11 107 219 191 63 197 98 171)(12 108 220 192 64 198 85 172)(13 109 221 193 65 199 86 173)(14 110 222 194 66 200 87 174)(15 133 145 115 47 75 155 42)(16 134 146 116 48 76 156 29)(17 135 147 117 49 77 157 30)(18 136 148 118 50 78 158 31)(19 137 149 119 51 79 159 32)(20 138 150 120 52 80 160 33)(21 139 151 121 53 81 161 34)(22 140 152 122 54 82 162 35)(23 127 153 123 55 83 163 36)(24 128 154 124 56 84 164 37)(25 129 141 125 43 71 165 38)(26 130 142 126 44 72 166 39)(27 131 143 113 45 73 167 40)(28 132 144 114 46 74 168 41)
(29 116)(30 117)(31 118)(32 119)(33 120)(34 121)(35 122)(36 123)(37 124)(38 125)(39 126)(40 113)(41 114)(42 115)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 127)(84 128)(99 203)(100 204)(101 205)(102 206)(103 207)(104 208)(105 209)(106 210)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)(169 189)(170 190)(171 191)(172 192)(173 193)(174 194)(175 195)(176 196)(177 183)(178 184)(179 185)(180 186)(181 187)(182 188)
(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 55 8 48)(2 54 9 47)(3 53 10 46)(4 52 11 45)(5 51 12 44)(6 50 13 43)(7 49 14 56)(15 68 22 61)(16 67 23 60)(17 66 24 59)(18 65 25 58)(19 64 26 57)(20 63 27 70)(21 62 28 69)(29 195 36 188)(30 194 37 187)(31 193 38 186)(32 192 39 185)(33 191 40 184)(34 190 41 183)(35 189 42 196)(71 206 78 199)(72 205 79 198)(73 204 80 197)(74 203 81 210)(75 202 82 209)(76 201 83 208)(77 200 84 207)(85 166 92 159)(86 165 93 158)(87 164 94 157)(88 163 95 156)(89 162 96 155)(90 161 97 168)(91 160 98 167)(99 139 106 132)(100 138 107 131)(101 137 108 130)(102 136 109 129)(103 135 110 128)(104 134 111 127)(105 133 112 140)(113 178 120 171)(114 177 121 170)(115 176 122 169)(116 175 123 182)(117 174 124 181)(118 173 125 180)(119 172 126 179)(141 214 148 221)(142 213 149 220)(143 212 150 219)(144 211 151 218)(145 224 152 217)(146 223 153 216)(147 222 154 215)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 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 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 2 2 2 2 28 28 28 28 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - + - + + - + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 C4×D7 C4.D4 C4.10D4 C28.46D4 C4.12D28 kernel M4(2)⋊Dic7 C2×C4.Dic7 C2×C4⋊Dic7 C14×M4(2) C4.Dic7 C7×M4(2) C22×Dic7 C2×C28 C2×C28 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C14 C14 C2 C2 # reps 1 1 1 1 4 4 4 3 1 3 6 3 6 6 6 12 6 1 1 6 6

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

 98 0 0 0 0 0 108 15 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 112 0 0 0 0 1 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 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 83 0 0 0 0 0 22 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 99 84 0 0 0 0 112 14 0 0 0 0 0 0 18 109 0 0 0 0 109 95 0 0 0 0 0 0 18 109 0 0 0 0 109 95

G:=sub<GL(6,GF(113))| [98,108,0,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,112,0,0,0,1,0,0,0,0,0,0,1,0,0],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[83,22,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[99,112,0,0,0,0,84,14,0,0,0,0,0,0,18,109,0,0,0,0,109,95,0,0,0,0,0,0,18,109,0,0,0,0,109,95] >;

M4(2)⋊Dic7 in GAP, Magma, Sage, TeX

M_4(2)\rtimes {\rm Dic}_7
% in TeX

G:=Group("M4(2):Dic7");
// GroupNames label

G:=SmallGroup(448,111);
// by ID

G=gap.SmallGroup(448,111);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,1123,136,851,18822]);
// Polycyclic

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

׿
×
𝔽