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## G = (C7×Q8).D4order 448 = 26·7

### 6th non-split extension by C7×Q8 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C7×Q8).D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — Q8×Dic7 — (C7×Q8).D4
 Lower central C7 — C14 — C2×C28 — (C7×Q8).D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for (C7×Q8).D4
G = < a,b,c,d,e | a7=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 484 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C7⋊C8, C56, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×C14, Q8.D4, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4⋊Dic7, C4⋊Dic7, C7⋊Q16, C23.D7, C2×C56, C7×SD16, C2×Dic14, D4×C14, Q8×C14, Dic7⋊C8, C28.44D4, D4⋊Dic7, C28.17D4, C2×C7⋊Q16, Q8×Dic7, C14×SD16, (C7×Q8).D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8.C22, C7⋊D4, C22×D7, Q8.D4, D4×D7, D42D7, C2×C7⋊D4, SD16⋊D7, SD163D7, Dic7⋊D4, (C7×Q8).D4

Smallest permutation representation of (C7×Q8).D4
On 224 points
Generators in S224
(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 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)(113 148 120 141)(114 149 121 142)(115 150 122 143)(116 151 123 144)(117 152 124 145)(118 153 125 146)(119 154 126 147)(127 162 134 155)(128 163 135 156)(129 164 136 157)(130 165 137 158)(131 166 138 159)(132 167 139 160)(133 168 140 161)(169 197 176 204)(170 198 177 205)(171 199 178 206)(172 200 179 207)(173 201 180 208)(174 202 181 209)(175 203 182 210)(183 211 190 218)(184 212 191 219)(185 213 192 220)(186 214 193 221)(187 215 194 222)(188 216 195 223)(189 217 196 224)
(1 64 8 57)(2 65 9 58)(3 66 10 59)(4 67 11 60)(5 68 12 61)(6 69 13 62)(7 70 14 63)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)(113 176 120 169)(114 177 121 170)(115 178 122 171)(116 179 123 172)(117 180 124 173)(118 181 125 174)(119 182 126 175)(127 190 134 183)(128 191 135 184)(129 192 136 185)(130 193 137 186)(131 194 138 187)(132 195 139 188)(133 196 140 189)(141 204 148 197)(142 205 149 198)(143 206 150 199)(144 207 151 200)(145 208 152 201)(146 209 153 202)(147 210 154 203)(155 218 162 211)(156 219 163 212)(157 220 164 213)(158 221 165 214)(159 222 166 215)(160 223 167 216)(161 224 168 217)
(1 132 15 118)(2 131 16 117)(3 130 17 116)(4 129 18 115)(5 128 19 114)(6 127 20 113)(7 133 21 119)(8 139 22 125)(9 138 23 124)(10 137 24 123)(11 136 25 122)(12 135 26 121)(13 134 27 120)(14 140 28 126)(29 160 43 146)(30 159 44 145)(31 158 45 144)(32 157 46 143)(33 156 47 142)(34 155 48 141)(35 161 49 147)(36 167 50 153)(37 166 51 152)(38 165 52 151)(39 164 53 150)(40 163 54 149)(41 162 55 148)(42 168 56 154)(57 188 71 174)(58 187 72 173)(59 186 73 172)(60 185 74 171)(61 184 75 170)(62 183 76 169)(63 189 77 175)(64 195 78 181)(65 194 79 180)(66 193 80 179)(67 192 81 178)(68 191 82 177)(69 190 83 176)(70 196 84 182)(85 216 99 202)(86 215 100 201)(87 214 101 200)(88 213 102 199)(89 212 103 198)(90 211 104 197)(91 217 105 203)(92 223 106 209)(93 222 107 208)(94 221 108 207)(95 220 109 206)(96 219 110 205)(97 218 111 204)(98 224 112 210)
(1 125 8 118)(2 124 9 117)(3 123 10 116)(4 122 11 115)(5 121 12 114)(6 120 13 113)(7 126 14 119)(15 139 22 132)(16 138 23 131)(17 137 24 130)(18 136 25 129)(19 135 26 128)(20 134 27 127)(21 140 28 133)(29 146 36 153)(30 145 37 152)(31 144 38 151)(32 143 39 150)(33 142 40 149)(34 141 41 148)(35 147 42 154)(43 160 50 167)(44 159 51 166)(45 158 52 165)(46 157 53 164)(47 156 54 163)(48 155 55 162)(49 161 56 168)(57 202 64 209)(58 201 65 208)(59 200 66 207)(60 199 67 206)(61 198 68 205)(62 197 69 204)(63 203 70 210)(71 216 78 223)(72 215 79 222)(73 214 80 221)(74 213 81 220)(75 212 82 219)(76 211 83 218)(77 217 84 224)(85 174 92 181)(86 173 93 180)(87 172 94 179)(88 171 95 178)(89 170 96 177)(90 169 97 176)(91 175 98 182)(99 188 106 195)(100 187 107 194)(101 186 108 193)(102 185 109 192)(103 184 110 191)(104 183 111 190)(105 189 112 196)

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

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

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

61 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14O 28A ··· 28F 28G ··· 28L 56A ··· 56L order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 8 2 2 4 4 14 14 28 28 28 56 2 2 2 4 4 28 28 2 ··· 2 8 ··· 8 4 ··· 4 8 ··· 8 4 ··· 4

61 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D7 C4○D4 D14 D14 D14 C4○D8 C7⋊D4 C8.C22 D4⋊2D7 D4×D7 SD16⋊D7 SD16⋊3D7 kernel (C7×Q8).D4 Dic7⋊C8 C28.44D4 D4⋊Dic7 C28.17D4 C2×C7⋊Q16 Q8×Dic7 C14×SD16 C2×Dic7 C7×Q8 C2×SD16 C28 C2×C8 C2×D4 C2×Q8 C14 Q8 C14 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 3 2 3 3 3 4 12 1 3 3 6 6

Matrix representation of (C7×Q8).D4 in GL4(𝔽113) generated by

 49 0 0 0 1 30 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 73 0 0 17 112
,
 1 0 0 0 0 1 0 0 0 0 0 45 0 0 5 0
,
 88 23 0 0 17 25 0 0 0 0 98 0 0 0 0 98
,
 88 23 0 0 22 25 0 0 0 0 98 0 0 0 84 15
G:=sub<GL(4,GF(113))| [49,1,0,0,0,30,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,17,0,0,73,112],[1,0,0,0,0,1,0,0,0,0,0,5,0,0,45,0],[88,17,0,0,23,25,0,0,0,0,98,0,0,0,0,98],[88,22,0,0,23,25,0,0,0,0,98,84,0,0,0,15] >;

(C7×Q8).D4 in GAP, Magma, Sage, TeX

(C_7\times Q_8).D_4
% in TeX

G:=Group("(C7xQ8).D4");
// GroupNames label

G:=SmallGroup(448,700);
// by ID

G=gap.SmallGroup(448,700);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,1094,135,570,297,136,18822]);
// Polycyclic

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

׿
×
𝔽