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## G = C2×C28.C23order 448 = 26·7

### Direct product of C2 and C28.C23

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×C28.C23
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×C4○D28 — C2×C28.C23
 Lower central C7 — C14 — C28 — C2×C28.C23
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for C2×C28.C23
G = < a,b,c,d,e | a2=b28=c2=1, d2=e2=b14, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b15, cd=dc, ece-1=b21c, ede-1=b14d >

Subgroups: 1012 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C14, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×C14, C2×C8.C22, C2×C7⋊C8, C4.Dic7, Q8⋊D7, C7⋊Q16, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, Q8×C14, Q8×C14, C2×C4.Dic7, C2×Q8⋊D7, C28.C23, C2×C7⋊Q16, C2×C4○D28, Q8×C2×C14, C2×C28.C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, C7⋊D4, C22×D7, C2×C8.C22, C2×C7⋊D4, C23×D7, C28.C23, C22×C7⋊D4, C2×C28.C23

Smallest permutation representation of C2×C28.C23
On 224 points
Generators in S224
(1 80)(2 81)(3 82)(4 83)(5 84)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(113 195)(114 196)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 217)(142 218)(143 219)(144 220)(145 221)(146 222)(147 223)(148 224)(149 197)(150 198)(151 199)(152 200)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)
(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)
(1 136)(2 135)(3 134)(4 133)(5 132)(6 131)(7 130)(8 129)(9 128)(10 127)(11 126)(12 125)(13 124)(14 123)(15 122)(16 121)(17 120)(18 119)(19 118)(20 117)(21 116)(22 115)(23 114)(24 113)(25 140)(26 139)(27 138)(28 137)(29 162)(30 161)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 150)(42 149)(43 148)(44 147)(45 146)(46 145)(47 144)(48 143)(49 142)(50 141)(51 168)(52 167)(53 166)(54 165)(55 164)(56 163)(57 185)(58 184)(59 183)(60 182)(61 181)(62 180)(63 179)(64 178)(65 177)(66 176)(67 175)(68 174)(69 173)(70 172)(71 171)(72 170)(73 169)(74 196)(75 195)(76 194)(77 193)(78 192)(79 191)(80 190)(81 189)(82 188)(83 187)(84 186)(85 218)(86 217)(87 216)(88 215)(89 214)(90 213)(91 212)(92 211)(93 210)(94 209)(95 208)(96 207)(97 206)(98 205)(99 204)(100 203)(101 202)(102 201)(103 200)(104 199)(105 198)(106 197)(107 224)(108 223)(109 222)(110 221)(111 220)(112 219)
(1 8 15 22)(2 9 16 23)(3 10 17 24)(4 11 18 25)(5 12 19 26)(6 13 20 27)(7 14 21 28)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 134 127 120)(114 135 128 121)(115 136 129 122)(116 137 130 123)(117 138 131 124)(118 139 132 125)(119 140 133 126)(141 162 155 148)(142 163 156 149)(143 164 157 150)(144 165 158 151)(145 166 159 152)(146 167 160 153)(147 168 161 154)(169 190 183 176)(170 191 184 177)(171 192 185 178)(172 193 186 179)(173 194 187 180)(174 195 188 181)(175 196 189 182)(197 218 211 204)(198 219 212 205)(199 220 213 206)(200 221 214 207)(201 222 215 208)(202 223 216 209)(203 224 217 210)
(1 33 15 47)(2 48 16 34)(3 35 17 49)(4 50 18 36)(5 37 19 51)(6 52 20 38)(7 39 21 53)(8 54 22 40)(9 41 23 55)(10 56 24 42)(11 43 25 29)(12 30 26 44)(13 45 27 31)(14 32 28 46)(57 88 71 102)(58 103 72 89)(59 90 73 104)(60 105 74 91)(61 92 75 106)(62 107 76 93)(63 94 77 108)(64 109 78 95)(65 96 79 110)(66 111 80 97)(67 98 81 112)(68 85 82 99)(69 100 83 86)(70 87 84 101)(113 156 127 142)(114 143 128 157)(115 158 129 144)(116 145 130 159)(117 160 131 146)(118 147 132 161)(119 162 133 148)(120 149 134 163)(121 164 135 150)(122 151 136 165)(123 166 137 152)(124 153 138 167)(125 168 139 154)(126 155 140 141)(169 206 183 220)(170 221 184 207)(171 208 185 222)(172 223 186 209)(173 210 187 224)(174 197 188 211)(175 212 189 198)(176 199 190 213)(177 214 191 200)(178 201 192 215)(179 216 193 202)(180 203 194 217)(181 218 195 204)(182 205 196 219)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14U 28A ··· 28AJ order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 28 28 2 2 2 2 4 4 4 4 28 28 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 C7⋊D4 C7⋊D4 C8.C22 C28.C23 kernel C2×C28.C23 C2×C4.Dic7 C2×Q8⋊D7 C28.C23 C2×C7⋊Q16 C2×C4○D28 Q8×C2×C14 C2×C28 C22×C14 C22×Q8 C22×C4 C2×Q8 C2×C4 C23 C14 C2 # reps 1 1 2 8 2 1 1 3 1 3 3 18 18 6 2 12

Matrix representation of C2×C28.C23 in GL6(𝔽113)

 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 1 0 0 0 0 0 0 1
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 28 57 0 0 0 0 28 85 0 0 0 0 56 20 0 4 0 0 37 20 109 0
,
 93 22 0 0 0 0 100 20 0 0 0 0 0 0 80 0 111 0 0 0 0 0 112 1 0 0 92 0 33 0 0 0 92 1 33 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 2 0 0 0 0 112 1 0 0 0 0 0 80 0 112 0 0 33 80 1 0
,
 83 106 0 0 0 0 80 30 0 0 0 0 0 0 99 63 0 0 0 0 74 14 0 0 0 0 44 34 25 39 0 0 10 34 39 88

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,28,28,56,37,0,0,57,85,20,20,0,0,0,0,0,109,0,0,0,0,4,0],[93,100,0,0,0,0,22,20,0,0,0,0,0,0,80,0,92,92,0,0,0,0,0,1,0,0,111,112,33,33,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,112,0,33,0,0,2,1,80,80,0,0,0,0,0,1,0,0,0,0,112,0],[83,80,0,0,0,0,106,30,0,0,0,0,0,0,99,74,44,10,0,0,63,14,34,34,0,0,0,0,25,39,0,0,0,0,39,88] >;

C2×C28.C23 in GAP, Magma, Sage, TeX

C_2\times C_{28}.C_2^3
% in TeX

G:=Group("C2xC28.C2^3");
// GroupNames label

G:=SmallGroup(448,1261);
// by ID

G=gap.SmallGroup(448,1261);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,136,1684,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^28=c^2=1,d^2=e^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^15,c*d=d*c,e*c*e^-1=b^21*c,e*d*e^-1=b^14*d>;
// generators/relations

׿
×
𝔽