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## G = C23.49D28order 448 = 26·7

### 20th non-split extension by C23 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C23.49D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C2×C4○D28 — C23.49D28
 Lower central C7 — C14 — C28 — C23.49D28
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C23.49D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >

Subgroups: 868 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C7, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, D7 [×2], C14 [×3], C14 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic7 [×4], C28 [×2], C28 [×2], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C56 [×2], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7 [×5], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C22×D7, C22×C14, C23.36D4, C4⋊Dic7 [×2], C4⋊Dic7, C2×C56 [×2], C7×M4(2) [×2], C2×Dic14, C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C22×Dic7, C2×C7⋊D4, C22×C28, C28.44D4 [×2], C2.D56 [×2], C2×C4⋊Dic7, C14×M4(2), C2×C4○D28, C23.49D28
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D7, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D14 [×3], C2×C22⋊C4, C8⋊C22, C8.C22, C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C22×D7, C23.36D4, D14⋊C4 [×4], C2×C4×D7, C2×D28, C2×C7⋊D4, C8⋊D14, C8.D14, C2×D14⋊C4, C23.49D28

Smallest permutation representation of C23.49D28
On 224 points
Generators in S224
```(1 137)(2 166)(3 139)(4 168)(5 141)(6 114)(7 143)(8 116)(9 145)(10 118)(11 147)(12 120)(13 149)(14 122)(15 151)(16 124)(17 153)(18 126)(19 155)(20 128)(21 157)(22 130)(23 159)(24 132)(25 161)(26 134)(27 163)(28 136)(29 165)(30 138)(31 167)(32 140)(33 113)(34 142)(35 115)(36 144)(37 117)(38 146)(39 119)(40 148)(41 121)(42 150)(43 123)(44 152)(45 125)(46 154)(47 127)(48 156)(49 129)(50 158)(51 131)(52 160)(53 133)(54 162)(55 135)(56 164)(57 187)(58 216)(59 189)(60 218)(61 191)(62 220)(63 193)(64 222)(65 195)(66 224)(67 197)(68 170)(69 199)(70 172)(71 201)(72 174)(73 203)(74 176)(75 205)(76 178)(77 207)(78 180)(79 209)(80 182)(81 211)(82 184)(83 213)(84 186)(85 215)(86 188)(87 217)(88 190)(89 219)(90 192)(91 221)(92 194)(93 223)(94 196)(95 169)(96 198)(97 171)(98 200)(99 173)(100 202)(101 175)(102 204)(103 177)(104 206)(105 179)(106 208)(107 181)(108 210)(109 183)(110 212)(111 185)(112 214)
(1 137)(2 138)(3 139)(4 140)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 161)(26 162)(27 163)(28 164)(29 165)(30 166)(31 167)(32 168)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(65 195)(66 196)(67 197)(68 198)(69 199)(70 200)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 211)(82 212)(83 213)(84 214)(85 215)(86 216)(87 217)(88 218)(89 219)(90 220)(91 221)(92 222)(93 223)(94 224)(95 169)(96 170)(97 171)(98 172)(99 173)(100 174)(101 175)(102 176)(103 177)(104 178)(105 179)(106 180)(107 181)(108 182)(109 183)(110 184)(111 185)(112 186)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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)
(1 103 165 205)(2 204 166 102)(3 101 167 203)(4 202 168 100)(5 99 113 201)(6 200 114 98)(7 97 115 199)(8 198 116 96)(9 95 117 197)(10 196 118 94)(11 93 119 195)(12 194 120 92)(13 91 121 193)(14 192 122 90)(15 89 123 191)(16 190 124 88)(17 87 125 189)(18 188 126 86)(19 85 127 187)(20 186 128 84)(21 83 129 185)(22 184 130 82)(23 81 131 183)(24 182 132 80)(25 79 133 181)(26 180 134 78)(27 77 135 179)(28 178 136 76)(29 75 137 177)(30 176 138 74)(31 73 139 175)(32 174 140 72)(33 71 141 173)(34 172 142 70)(35 69 143 171)(36 170 144 68)(37 67 145 169)(38 224 146 66)(39 65 147 223)(40 222 148 64)(41 63 149 221)(42 220 150 62)(43 61 151 219)(44 218 152 60)(45 59 153 217)(46 216 154 58)(47 57 155 215)(48 214 156 112)(49 111 157 213)(50 212 158 110)(51 109 159 211)(52 210 160 108)(53 107 161 209)(54 208 162 106)(55 105 163 207)(56 206 164 104)```

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 2 2 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 2 2 28 28 2 2 2 2 28 ··· 28 2 2 2 4 4 4 4 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 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D7 D14 D14 C4×D7 D28 C7⋊D4 D28 C8⋊C22 C8.C22 C8⋊D14 C8.D14 kernel C23.49D28 C28.44D4 C2.D56 C2×C4⋊Dic7 C14×M4(2) C2×C4○D28 C4○D28 C2×C28 C22×C14 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C14 C14 C2 C2 # reps 1 2 2 1 1 1 8 3 1 3 6 3 12 6 12 6 1 1 6 6

Matrix representation of C23.49D28 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 18 0 1 0 0 0 18 0 0 1
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 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 0 112 0 0 0 0 0 0 112
,
 73 28 0 0 0 0 57 59 0 0 0 0 0 0 98 0 36 0 0 0 0 0 112 1 0 0 0 112 15 0 0 0 44 0 15 0
,
 86 5 0 0 0 0 80 27 0 0 0 0 0 0 69 66 0 0 0 0 58 44 0 0 0 0 80 84 17 86 0 0 0 84 86 96

`G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,18,18,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[73,57,0,0,0,0,28,59,0,0,0,0,0,0,98,0,0,44,0,0,0,0,112,0,0,0,36,112,15,15,0,0,0,1,0,0],[86,80,0,0,0,0,5,27,0,0,0,0,0,0,69,58,80,0,0,0,66,44,84,84,0,0,0,0,17,86,0,0,0,0,86,96] >;`

C23.49D28 in GAP, Magma, Sage, TeX

`C_2^3._{49}D_{28}`
`% in TeX`

`G:=Group("C2^3.49D28");`
`// GroupNames label`

`G:=SmallGroup(448,667);`
`// by ID`

`G=gap.SmallGroup(448,667);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,387,142,1123,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;`
`// generators/relations`

׿
×
𝔽