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## G = C4○D4×Dic7order 448 = 26·7

### Direct product of C4○D4 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4○D4×Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×C4×Dic7 — C4○D4×Dic7
 Lower central C7 — C14 — C4○D4×Dic7
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C4○D4×Dic7
G = < a,b,c,d,e | a4=c2=d14=1, b2=a2, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 916 in 310 conjugacy classes, 195 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C4×C4○D4, C4×Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C2×C4×Dic7, C23.21D14, D4×Dic7, Q8×Dic7, C14×C4○D4, C4○D4×Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, Dic7, D14, C23×C4, C2×C4○D4, C2×Dic7, C22×D7, C4×C4○D4, C22×Dic7, C23×D7, D7×C4○D4, C23×Dic7, C4○D4×Dic7

Smallest permutation representation of C4○D4×Dic7
On 224 points
Generators in S224
(1 180 124 45)(2 181 125 46)(3 182 126 47)(4 169 113 48)(5 170 114 49)(6 171 115 50)(7 172 116 51)(8 173 117 52)(9 174 118 53)(10 175 119 54)(11 176 120 55)(12 177 121 56)(13 178 122 43)(14 179 123 44)(15 94 193 67)(16 95 194 68)(17 96 195 69)(18 97 196 70)(19 98 183 57)(20 85 184 58)(21 86 185 59)(22 87 186 60)(23 88 187 61)(24 89 188 62)(25 90 189 63)(26 91 190 64)(27 92 191 65)(28 93 192 66)(29 222 198 160)(30 223 199 161)(31 224 200 162)(32 211 201 163)(33 212 202 164)(34 213 203 165)(35 214 204 166)(36 215 205 167)(37 216 206 168)(38 217 207 155)(39 218 208 156)(40 219 209 157)(41 220 210 158)(42 221 197 159)(71 136 143 108)(72 137 144 109)(73 138 145 110)(74 139 146 111)(75 140 147 112)(76 127 148 99)(77 128 149 100)(78 129 150 101)(79 130 151 102)(80 131 152 103)(81 132 153 104)(82 133 154 105)(83 134 141 106)(84 135 142 107)
(1 180 124 45)(2 181 125 46)(3 182 126 47)(4 169 113 48)(5 170 114 49)(6 171 115 50)(7 172 116 51)(8 173 117 52)(9 174 118 53)(10 175 119 54)(11 176 120 55)(12 177 121 56)(13 178 122 43)(14 179 123 44)(15 94 193 67)(16 95 194 68)(17 96 195 69)(18 97 196 70)(19 98 183 57)(20 85 184 58)(21 86 185 59)(22 87 186 60)(23 88 187 61)(24 89 188 62)(25 90 189 63)(26 91 190 64)(27 92 191 65)(28 93 192 66)(29 160 198 222)(30 161 199 223)(31 162 200 224)(32 163 201 211)(33 164 202 212)(34 165 203 213)(35 166 204 214)(36 167 205 215)(37 168 206 216)(38 155 207 217)(39 156 208 218)(40 157 209 219)(41 158 210 220)(42 159 197 221)(71 108 143 136)(72 109 144 137)(73 110 145 138)(74 111 146 139)(75 112 147 140)(76 99 148 127)(77 100 149 128)(78 101 150 129)(79 102 151 130)(80 103 152 131)(81 104 153 132)(82 105 154 133)(83 106 141 134)(84 107 142 135)
(1 82)(2 83)(3 84)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 157)(16 158)(17 159)(18 160)(19 161)(20 162)(21 163)(22 164)(23 165)(24 166)(25 167)(26 168)(27 155)(28 156)(29 97)(30 98)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 99)(54 100)(55 101)(56 102)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 197)(70 198)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 151)(122 152)(123 153)(124 154)(125 141)(126 142)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 181)(135 182)(136 169)(137 170)(138 171)(139 172)(140 173)(183 223)(184 224)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)
(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 23 8 16)(2 22 9 15)(3 21 10 28)(4 20 11 27)(5 19 12 26)(6 18 13 25)(7 17 14 24)(29 131 36 138)(30 130 37 137)(31 129 38 136)(32 128 39 135)(33 127 40 134)(34 140 41 133)(35 139 42 132)(43 63 50 70)(44 62 51 69)(45 61 52 68)(46 60 53 67)(47 59 54 66)(48 58 55 65)(49 57 56 64)(71 162 78 155)(72 161 79 168)(73 160 80 167)(74 159 81 166)(75 158 82 165)(76 157 83 164)(77 156 84 163)(85 176 92 169)(86 175 93 182)(87 174 94 181)(88 173 95 180)(89 172 96 179)(90 171 97 178)(91 170 98 177)(99 209 106 202)(100 208 107 201)(101 207 108 200)(102 206 109 199)(103 205 110 198)(104 204 111 197)(105 203 112 210)(113 184 120 191)(114 183 121 190)(115 196 122 189)(116 195 123 188)(117 194 124 187)(118 193 125 186)(119 192 126 185)(141 212 148 219)(142 211 149 218)(143 224 150 217)(144 223 151 216)(145 222 152 215)(146 221 153 214)(147 220 154 213)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 4S ··· 4AD 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 7 ··· 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D7 C4○D4 D14 D14 D14 Dic7 D7×C4○D4 kernel C4○D4×Dic7 C2×C4×Dic7 C23.21D14 D4×Dic7 Q8×Dic7 C14×C4○D4 C7×C4○D4 C2×C4○D4 Dic7 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 3 3 6 2 1 16 3 8 9 9 3 24 12

Matrix representation of C4○D4×Dic7 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 12 0 0 0 0 12
,
 28 0 0 0 0 28 0 0 0 0 12 17 0 0 0 17
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 27 1
,
 8 28 0 0 6 3 0 0 0 0 1 0 0 0 0 1
,
 9 19 0 0 14 20 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,17,17],[28,0,0,0,0,28,0,0,0,0,28,27,0,0,0,1],[8,6,0,0,28,3,0,0,0,0,1,0,0,0,0,1],[9,14,0,0,19,20,0,0,0,0,28,0,0,0,0,28] >;

C4○D4×Dic7 in GAP, Magma, Sage, TeX

C_4\circ D_4\times {\rm Dic}_7
% in TeX

G:=Group("C4oD4xDic7");
// GroupNames label

G:=SmallGroup(448,1279);
// by ID

G=gap.SmallGroup(448,1279);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,570,18822]);
// Polycyclic

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

׿
×
𝔽