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## G = C4×Q8⋊D7order 448 = 26·7

### Direct product of C4 and Q8⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C4×Q8⋊D7
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C2×D28 — C2×Q8⋊D7 — C4×Q8⋊D7
 Lower central C7 — C14 — C28 — C4×Q8⋊D7
 Upper central C1 — C2×C4 — C42 — C4×Q8

Generators and relations for C4×Q8⋊D7
G = < a,b,c,d,e | a4=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 580 in 122 conjugacy classes, 55 normal (39 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, D14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.Q8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C7⋊C8, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C4×SD16, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, Q8⋊D7, C4×C28, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, Q8×C14, C4×C7⋊C8, C4.Dic14, C14.D8, Q8⋊Dic7, C4×D28, C2×Q8⋊D7, Q8×C28, C4×Q8⋊D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, SD16, C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×SD16, C4○D8, C4×D7, C7⋊D4, C22×D7, C4×SD16, Q8⋊D7, C2×C4×D7, C4○D28, C2×C7⋊D4, C4×C7⋊D4, C2×Q8⋊D7, D4.8D14, C4×Q8⋊D7

Smallest permutation representation of C4×Q8⋊D7
On 224 points
Generators in S224
(1 85 29 57)(2 86 30 58)(3 87 31 59)(4 88 32 60)(5 89 33 61)(6 90 34 62)(7 91 35 63)(8 92 36 64)(9 93 37 65)(10 94 38 66)(11 95 39 67)(12 96 40 68)(13 97 41 69)(14 98 42 70)(15 99 43 71)(16 100 44 72)(17 101 45 73)(18 102 46 74)(19 103 47 75)(20 104 48 76)(21 105 49 77)(22 106 50 78)(23 107 51 79)(24 108 52 80)(25 109 53 81)(26 110 54 82)(27 111 55 83)(28 112 56 84)(113 197 141 169)(114 198 142 170)(115 199 143 171)(116 200 144 172)(117 201 145 173)(118 202 146 174)(119 203 147 175)(120 204 148 176)(121 205 149 177)(122 206 150 178)(123 207 151 179)(124 208 152 180)(125 209 153 181)(126 210 154 182)(127 211 155 183)(128 212 156 184)(129 213 157 185)(130 214 158 186)(131 215 159 187)(132 216 160 188)(133 217 161 189)(134 218 162 190)(135 219 163 191)(136 220 164 192)(137 221 165 193)(138 222 166 194)(139 223 167 195)(140 224 168 196)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 148 8 141)(2 149 9 142)(3 150 10 143)(4 151 11 144)(5 152 12 145)(6 153 13 146)(7 154 14 147)(15 162 22 155)(16 163 23 156)(17 164 24 157)(18 165 25 158)(19 166 26 159)(20 167 27 160)(21 168 28 161)(29 120 36 113)(30 121 37 114)(31 122 38 115)(32 123 39 116)(33 124 40 117)(34 125 41 118)(35 126 42 119)(43 134 50 127)(44 135 51 128)(45 136 52 129)(46 137 53 130)(47 138 54 131)(48 139 55 132)(49 140 56 133)(57 204 64 197)(58 205 65 198)(59 206 66 199)(60 207 67 200)(61 208 68 201)(62 209 69 202)(63 210 70 203)(71 218 78 211)(72 219 79 212)(73 220 80 213)(74 221 81 214)(75 222 82 215)(76 223 83 216)(77 224 84 217)(85 176 92 169)(86 177 93 170)(87 178 94 171)(88 179 95 172)(89 180 96 173)(90 181 97 174)(91 182 98 175)(99 190 106 183)(100 191 107 184)(101 192 108 185)(102 193 109 186)(103 194 110 187)(104 195 111 188)(105 196 112 189)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 48)(24 47)(25 46)(26 45)(27 44)(28 43)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 112)(72 111)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(113 161)(114 160)(115 159)(116 158)(117 157)(118 156)(119 155)(120 168)(121 167)(122 166)(123 165)(124 164)(125 163)(126 162)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(169 217)(170 216)(171 215)(172 214)(173 213)(174 212)(175 211)(176 224)(177 223)(178 222)(179 221)(180 220)(181 219)(182 218)(183 203)(184 202)(185 201)(186 200)(187 199)(188 198)(189 197)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 7A 7B 7C 8A ··· 8H 14A ··· 14I 28A ··· 28L 28M ··· 28AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 28 28 1 1 1 1 2 2 2 2 4 4 4 4 28 28 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D7 SD16 C4○D4 D14 D14 D14 C4○D8 C7⋊D4 C4×D7 C4○D28 Q8⋊D7 D4.8D14 kernel C4×Q8⋊D7 C4×C7⋊C8 C4.Dic14 C14.D8 Q8⋊Dic7 C4×D28 C2×Q8⋊D7 Q8×C28 Q8⋊D7 C2×C28 C4×Q8 C28 C28 C42 C4⋊C4 C2×Q8 C14 C2×C4 Q8 C4 C4 C2 # reps 1 1 1 1 1 1 1 1 8 2 3 4 2 3 3 3 4 12 12 12 6 6

Matrix representation of C4×Q8⋊D7 in GL4(𝔽113) generated by

 98 0 0 0 0 98 0 0 0 0 112 0 0 0 0 112
,
 112 0 0 0 0 112 0 0 0 0 112 44 0 0 77 1
,
 27 5 0 0 103 86 0 0 0 0 87 7 0 0 97 26
,
 33 112 0 0 2 89 0 0 0 0 1 0 0 0 0 1
,
 81 80 0 0 31 32 0 0 0 0 1 69 0 0 0 112
G:=sub<GL(4,GF(113))| [98,0,0,0,0,98,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,112,77,0,0,44,1],[27,103,0,0,5,86,0,0,0,0,87,97,0,0,7,26],[33,2,0,0,112,89,0,0,0,0,1,0,0,0,0,1],[81,31,0,0,80,32,0,0,0,0,1,0,0,0,69,112] >;

C4×Q8⋊D7 in GAP, Magma, Sage, TeX

C_4\times Q_8\rtimes D_7
% in TeX

G:=Group("C4xQ8:D7");
// GroupNames label

G:=SmallGroup(448,559);
// by ID

G=gap.SmallGroup(448,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,232,58,1684,851,102,18822]);
// Polycyclic

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

׿
×
𝔽