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## G = C28⋊7M4(2)  order 448 = 26·7

### 1st semidirect product of C28 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28⋊7M4(2)
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C28⋊C8 — C28⋊7M4(2)
 Lower central C7 — C2×C14 — C28⋊7M4(2)
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C287M4(2)
G = < a,b,c | a28=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >

Subgroups: 324 in 126 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊C8, C2×C42, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C4⋊M4(2), C2×C7⋊C8, C4.Dic7, C4×C28, C4×C28, C22×C28, C22×C28, C28⋊C8, C2×C4.Dic7, C2×C4×C28, C287M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, C2×M4(2), Dic14, D28, C2×Dic7, C22×D7, C4⋊M4(2), C4.Dic7, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C2×C4.Dic7, C2×C4⋊Dic7, C287M4(2)

Smallest permutation representation of C287M4(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)
(1 212 43 104 171 155 81 137)(2 211 44 103 172 154 82 136)(3 210 45 102 173 153 83 135)(4 209 46 101 174 152 84 134)(5 208 47 100 175 151 57 133)(6 207 48 99 176 150 58 132)(7 206 49 98 177 149 59 131)(8 205 50 97 178 148 60 130)(9 204 51 96 179 147 61 129)(10 203 52 95 180 146 62 128)(11 202 53 94 181 145 63 127)(12 201 54 93 182 144 64 126)(13 200 55 92 183 143 65 125)(14 199 56 91 184 142 66 124)(15 198 29 90 185 141 67 123)(16 197 30 89 186 168 68 122)(17 224 31 88 187 167 69 121)(18 223 32 87 188 166 70 120)(19 222 33 86 189 165 71 119)(20 221 34 85 190 164 72 118)(21 220 35 112 191 163 73 117)(22 219 36 111 192 162 74 116)(23 218 37 110 193 161 75 115)(24 217 38 109 194 160 76 114)(25 216 39 108 195 159 77 113)(26 215 40 107 196 158 78 140)(27 214 41 106 169 157 79 139)(28 213 42 105 170 156 80 138)
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 191)(8 192)(9 193)(10 194)(11 195)(12 196)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 81)(30 82)(31 83)(32 84)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)```

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28BT order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 2 2 2 28 ··· 28 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + - + - + - + image C1 C2 C2 C2 C4 C4 D4 Q8 D7 M4(2) Dic7 D14 Dic7 D14 Dic14 D28 C4.Dic7 kernel C28⋊7M4(2) C28⋊C8 C2×C4.Dic7 C2×C4×C28 C4×C28 C22×C28 C2×C28 C2×C28 C2×C42 C28 C42 C42 C22×C4 C22×C4 C2×C4 C2×C4 C4 # reps 1 4 2 1 4 4 2 2 3 8 6 6 6 3 12 12 48

Matrix representation of C287M4(2) in GL4(𝔽113) generated by

 57 8 0 0 0 2 0 0 0 0 109 0 0 0 71 28
,
 47 69 0 0 30 66 0 0 0 0 62 55 0 0 29 51
,
 1 27 0 0 0 112 0 0 0 0 112 0 0 0 68 1
`G:=sub<GL(4,GF(113))| [57,0,0,0,8,2,0,0,0,0,109,71,0,0,0,28],[47,30,0,0,69,66,0,0,0,0,62,29,0,0,55,51],[1,0,0,0,27,112,0,0,0,0,112,68,0,0,0,1] >;`

C287M4(2) in GAP, Magma, Sage, TeX

`C_{28}\rtimes_7M_4(2)`
`% in TeX`

`G:=Group("C28:7M4(2)");`
`// GroupNames label`

`G:=SmallGroup(448,458);`
`// by ID`

`G=gap.SmallGroup(448,458);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,253,120,758,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;`
`// generators/relations`

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