Copied to
clipboard

## G = C2×D14.5D4order 448 = 26·7

### Direct product of C2 and D14.5D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×D14.5D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — C2×D14.5D4
 Lower central C7 — C2×C14 — C2×D14.5D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×D14.5D4
G = < a,b,c,d,e | a2=b14=c2=d4=1, e2=b7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b7c, ede-1=d-1 >

Subgroups: 1796 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C2×C22.D4, Dic7⋊C4, D14⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C22×Dic7, C22×C28, C23×D7, D14.5D4, C2×Dic7⋊C4, C2×D14⋊C4, C14×C4⋊C4, D7×C22×C4, C22×D28, C2×D14.5D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, C22×D7, C2×C22.D4, C4○D28, D4×D7, Q82D7, C23×D7, D14.5D4, C2×C4○D28, C2×D4×D7, C2×Q82D7, C2×D14.5D4

Smallest permutation representation of C2×D14.5D4
On 224 points
Generators in S224
(1 66)(2 67)(3 68)(4 69)(5 70)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 126)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 121)(25 122)(26 123)(27 124)(28 125)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 181)(36 182)(37 169)(38 170)(39 171)(40 172)(41 173)(42 174)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 99)(55 100)(56 101)(71 201)(72 202)(73 203)(74 204)(75 205)(76 206)(77 207)(78 208)(79 209)(80 210)(81 197)(82 198)(83 199)(84 200)(85 213)(86 214)(87 215)(88 216)(89 217)(90 218)(91 219)(92 220)(93 221)(94 222)(95 223)(96 224)(97 211)(98 212)(127 167)(128 168)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 195)(142 196)(143 183)(144 184)(145 185)(146 186)(147 187)(148 188)(149 189)(150 190)(151 191)(152 192)(153 193)(154 194)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 70)(11 69)(12 68)(13 67)(14 66)(15 114)(16 113)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 182)(30 181)(31 180)(32 179)(33 178)(34 177)(35 176)(36 175)(37 174)(38 173)(39 172)(40 171)(41 170)(42 169)(43 99)(44 112)(45 111)(46 110)(47 109)(48 108)(49 107)(50 106)(51 105)(52 104)(53 103)(54 102)(55 101)(56 100)(71 207)(72 206)(73 205)(74 204)(75 203)(76 202)(77 201)(78 200)(79 199)(80 198)(81 197)(82 210)(83 209)(84 208)(85 213)(86 212)(87 211)(88 224)(89 223)(90 222)(91 221)(92 220)(93 219)(94 218)(95 217)(96 216)(97 215)(98 214)(127 155)(128 168)(129 167)(130 166)(131 165)(132 164)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 156)(141 192)(142 191)(143 190)(144 189)(145 188)(146 187)(147 186)(148 185)(149 184)(150 183)(151 196)(152 195)(153 194)(154 193)
(1 132 33 27)(2 133 34 28)(3 134 35 15)(4 135 36 16)(5 136 37 17)(6 137 38 18)(7 138 39 19)(8 139 40 20)(9 140 41 21)(10 127 42 22)(11 128 29 23)(12 129 30 24)(13 130 31 25)(14 131 32 26)(43 202 141 218)(44 203 142 219)(45 204 143 220)(46 205 144 221)(47 206 145 222)(48 207 146 223)(49 208 147 224)(50 209 148 211)(51 210 149 212)(52 197 150 213)(53 198 151 214)(54 199 152 215)(55 200 153 216)(56 201 154 217)(57 163 170 115)(58 164 171 116)(59 165 172 117)(60 166 173 118)(61 167 174 119)(62 168 175 120)(63 155 176 121)(64 156 177 122)(65 157 178 123)(66 158 179 124)(67 159 180 125)(68 160 181 126)(69 161 182 113)(70 162 169 114)(71 194 89 101)(72 195 90 102)(73 196 91 103)(74 183 92 104)(75 184 93 105)(76 185 94 106)(77 186 95 107)(78 187 96 108)(79 188 97 109)(80 189 98 110)(81 190 85 111)(82 191 86 112)(83 192 87 99)(84 193 88 100)
(1 208 8 201)(2 209 9 202)(3 210 10 203)(4 197 11 204)(5 198 12 205)(6 199 13 206)(7 200 14 207)(15 149 22 142)(16 150 23 143)(17 151 24 144)(18 152 25 145)(19 153 26 146)(20 154 27 147)(21 141 28 148)(29 220 36 213)(30 221 37 214)(31 222 38 215)(32 223 39 216)(33 224 40 217)(34 211 41 218)(35 212 42 219)(43 133 50 140)(44 134 51 127)(45 135 52 128)(46 136 53 129)(47 137 54 130)(48 138 55 131)(49 139 56 132)(57 83 64 76)(58 84 65 77)(59 71 66 78)(60 72 67 79)(61 73 68 80)(62 74 69 81)(63 75 70 82)(85 175 92 182)(86 176 93 169)(87 177 94 170)(88 178 95 171)(89 179 96 172)(90 180 97 173)(91 181 98 174)(99 156 106 163)(100 157 107 164)(101 158 108 165)(102 159 109 166)(103 160 110 167)(104 161 111 168)(105 162 112 155)(113 190 120 183)(114 191 121 184)(115 192 122 185)(116 193 123 186)(117 194 124 187)(118 195 125 188)(119 196 126 189)

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

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

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

88 conjugacy classes

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

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 C4○D28 D4×D7 Q8⋊2D7 kernel C2×D14.5D4 D14.5D4 C2×Dic7⋊C4 C2×D14⋊C4 C14×C4⋊C4 D7×C22×C4 C22×D28 C22×D7 C2×C4⋊C4 C2×C14 C4⋊C4 C22×C4 C22 C22 C22 # reps 1 8 1 3 1 1 1 4 3 8 12 9 24 6 6

Matrix representation of C2×D14.5D4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 0 4 0 0 0 0 7 7 0 0 0 0 0 0 17 25 0 0 0 0 8 5 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 1 0 0 0 0 0 28 0 0 0 0 0 0 7 28 0 0 0 0 19 22 0 0 0 0 0 0 1 0 0 0 0 0 13 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 26 2 0 0 0 0 25 3 0 0 0 0 0 0 1 11 0 0 0 0 13 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 5 0 0 0 0 19 22 0 0 0 0 0 0 17 13 0 0 0 0 0 12

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,7,0,0,0,0,4,7,0,0,0,0,0,0,17,8,0,0,0,0,25,5,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,1,28,0,0,0,0,0,0,7,19,0,0,0,0,28,22,0,0,0,0,0,0,1,13,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,25,0,0,0,0,2,3,0,0,0,0,0,0,1,13,0,0,0,0,11,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,19,0,0,0,0,5,22,0,0,0,0,0,0,17,0,0,0,0,0,13,12] >;

C2×D14.5D4 in GAP, Magma, Sage, TeX

C_2\times D_{14}._5D_4
% in TeX

G:=Group("C2xD14.5D4");
// GroupNames label

G:=SmallGroup(448,958);
// by ID

G=gap.SmallGroup(448,958);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,675,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^4=1,e^2=b^7,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,b*e=e*b,d*c*d^-1=e*c*e^-1=b^7*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽