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G = C7×Q8○D8order 448 = 26·7

Direct product of C7 and Q8○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C7×Q8○D8
 Chief series C1 — C2 — C4 — C28 — C7×Q8 — C7×Q16 — C14×Q16 — C7×Q8○D8
 Lower central C1 — C2 — C4 — C7×Q8○D8
 Upper central C1 — C14 — C7×C4○D4 — C7×Q8○D8

Generators and relations for C7×Q8○D8
G = < a,b,c,d,e | a7=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 346 in 248 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, Q8○D8, C2×C56, C7×M4(2), C7×D8, C7×SD16, C7×Q16, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C7×C8○D4, C14×Q16, C7×C4○D8, C7×C8.C22, C7×2- 1+4, C7×Q8○D8
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C24, C2×C14, C22×D4, C7×D4, C22×C14, Q8○D8, D4×C14, C23×C14, D4×C2×C14, C7×Q8○D8

Smallest permutation representation of C7×Q8○D8
On 224 points
Generators in S224
(1 125 46 66 201 117 38)(2 126 47 67 202 118 39)(3 127 48 68 203 119 40)(4 128 41 69 204 120 33)(5 121 42 70 205 113 34)(6 122 43 71 206 114 35)(7 123 44 72 207 115 36)(8 124 45 65 208 116 37)(9 148 164 89 156 180 97)(10 149 165 90 157 181 98)(11 150 166 91 158 182 99)(12 151 167 92 159 183 100)(13 152 168 93 160 184 101)(14 145 161 94 153 177 102)(15 146 162 95 154 178 103)(16 147 163 96 155 179 104)(17 196 141 25 83 188 105)(18 197 142 26 84 189 106)(19 198 143 27 85 190 107)(20 199 144 28 86 191 108)(21 200 137 29 87 192 109)(22 193 138 30 88 185 110)(23 194 139 31 81 186 111)(24 195 140 32 82 187 112)(49 77 212 174 61 220 136)(50 78 213 175 62 221 129)(51 79 214 176 63 222 130)(52 80 215 169 64 223 131)(53 73 216 170 57 224 132)(54 74 209 171 58 217 133)(55 75 210 172 59 218 134)(56 76 211 173 60 219 135)
(1 211 5 215)(2 212 6 216)(3 213 7 209)(4 214 8 210)(9 85 13 81)(10 86 14 82)(11 87 15 83)(12 88 16 84)(17 91 21 95)(18 92 22 96)(19 93 23 89)(20 94 24 90)(25 99 29 103)(26 100 30 104)(27 101 31 97)(28 102 32 98)(33 79 37 75)(34 80 38 76)(35 73 39 77)(36 74 40 78)(41 63 45 59)(42 64 46 60)(43 57 47 61)(44 58 48 62)(49 114 53 118)(50 115 54 119)(51 116 55 120)(52 117 56 113)(65 218 69 222)(66 219 70 223)(67 220 71 224)(68 221 72 217)(105 166 109 162)(106 167 110 163)(107 168 111 164)(108 161 112 165)(121 169 125 173)(122 170 126 174)(123 171 127 175)(124 172 128 176)(129 207 133 203)(130 208 134 204)(131 201 135 205)(132 202 136 206)(137 178 141 182)(138 179 142 183)(139 180 143 184)(140 181 144 177)(145 187 149 191)(146 188 150 192)(147 189 151 185)(148 190 152 186)(153 195 157 199)(154 196 158 200)(155 197 159 193)(156 198 160 194)
(1 209 5 213)(2 210 6 214)(3 211 7 215)(4 212 8 216)(9 83 13 87)(10 84 14 88)(11 85 15 81)(12 86 16 82)(17 93 21 89)(18 94 22 90)(19 95 23 91)(20 96 24 92)(25 101 29 97)(26 102 30 98)(27 103 31 99)(28 104 32 100)(33 77 37 73)(34 78 38 74)(35 79 39 75)(36 80 40 76)(41 61 45 57)(42 62 46 58)(43 63 47 59)(44 64 48 60)(49 116 53 120)(50 117 54 113)(51 118 55 114)(52 119 56 115)(65 224 69 220)(66 217 70 221)(67 218 71 222)(68 219 72 223)(105 168 109 164)(106 161 110 165)(107 162 111 166)(108 163 112 167)(121 175 125 171)(122 176 126 172)(123 169 127 173)(124 170 128 174)(129 201 133 205)(130 202 134 206)(131 203 135 207)(132 204 136 208)(137 180 141 184)(138 181 142 177)(139 182 143 178)(140 183 144 179)(145 185 149 189)(146 186 150 190)(147 187 151 191)(148 188 152 192)(153 193 157 197)(154 194 158 198)(155 195 159 199)(156 196 160 200)
(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 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 130)(10 129)(11 136)(12 135)(13 134)(14 133)(15 132)(16 131)(25 67)(26 66)(27 65)(28 72)(29 71)(30 70)(31 69)(32 68)(33 111)(34 110)(35 109)(36 108)(37 107)(38 106)(39 105)(40 112)(41 139)(42 138)(43 137)(44 144)(45 143)(46 142)(47 141)(48 140)(49 150)(50 149)(51 148)(52 147)(53 146)(54 145)(55 152)(56 151)(57 178)(58 177)(59 184)(60 183)(61 182)(62 181)(63 180)(64 179)(73 162)(74 161)(75 168)(76 167)(77 166)(78 165)(79 164)(80 163)(81 204)(82 203)(83 202)(84 201)(85 208)(86 207)(87 206)(88 205)(89 214)(90 213)(91 212)(92 211)(93 210)(94 209)(95 216)(96 215)(97 222)(98 221)(99 220)(100 219)(101 218)(102 217)(103 224)(104 223)(113 185)(114 192)(115 191)(116 190)(117 189)(118 188)(119 187)(120 186)(121 193)(122 200)(123 199)(124 198)(125 197)(126 196)(127 195)(128 194)(153 171)(154 170)(155 169)(156 176)(157 175)(158 174)(159 173)(160 172)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E ··· 4J 7A ··· 7F 8A 8B 8C 8D 8E 14A ··· 14F 14G ··· 14X 14Y ··· 14AJ 28A ··· 28X 28Y ··· 28BH 56A ··· 56L 56M ··· 56AD order 1 2 2 2 2 2 2 4 4 4 4 4 ··· 4 7 ··· 7 8 8 8 8 8 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 56 ··· 56 size 1 1 2 2 2 4 4 2 2 2 2 4 ··· 4 1 ··· 1 2 2 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 C7×D4 C7×D4 Q8○D8 C7×Q8○D8 kernel C7×Q8○D8 C7×C8○D4 C14×Q16 C7×C4○D8 C7×C8.C22 C7×2- 1+4 Q8○D8 C8○D4 C2×Q16 C4○D8 C8.C22 2- 1+4 C7×D4 C7×Q8 D4 Q8 C7 C1 # reps 1 1 3 3 6 2 6 6 18 18 36 12 3 1 18 6 2 12

Matrix representation of C7×Q8○D8 in GL4(𝔽113) generated by

 109 0 0 0 0 109 0 0 0 0 109 0 0 0 0 109
,
 21 21 6 5 0 0 112 0 0 1 0 0 2 1 92 92
,
 92 0 108 108 0 0 0 112 111 112 21 21 0 1 0 0
,
 0 31 0 27 51 51 86 86 0 0 82 82 0 0 31 82
,
 99 30 72 25 0 0 84 47 19 95 83 83 55 37 44 44
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[21,0,0,2,21,0,1,1,6,112,0,92,5,0,0,92],[92,0,111,0,0,0,112,1,108,0,21,0,108,112,21,0],[0,51,0,0,31,51,0,0,0,86,82,31,27,86,82,82],[99,0,19,55,30,0,95,37,72,84,83,44,25,47,83,44] >;

C7×Q8○D8 in GAP, Magma, Sage, TeX

C_7\times Q_8\circ D_8
% in TeX

G:=Group("C7xQ8oD8");
// GroupNames label

G:=SmallGroup(448,1361);
// by ID

G=gap.SmallGroup(448,1361);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1576,1641,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations

׿
×
𝔽