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## G = C7×C22.C42order 448 = 26·7

### Direct product of C7 and C22.C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C22.C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C28 — C14×M4(2) — C7×C22.C42
 Lower central C1 — C2 — C22 — C7×C22.C42
 Upper central C1 — C2×C14 — C22×C28 — C7×C22.C42

Generators and relations for C7×C22.C42
G = < a,b,c,d,e | a7=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 154 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C4⋊C4, C2×M4(2), C56, C2×C28, C2×C28, C2×C28, C22×C14, C22.C42, C7×C4⋊C4, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C28, C14×C4⋊C4, C14×M4(2), C7×C22.C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C4.D4, C4.10D4, C2×C28, C7×D4, C7×Q8, C22.C42, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C4.D4, C7×C4.10D4, C7×C22.C42

Smallest permutation representation of C7×C22.C42
On 224 points
Generators in S224
(1 45 151 39 143 31 135)(2 46 152 40 144 32 136)(3 47 145 33 137 25 129)(4 48 146 34 138 26 130)(5 41 147 35 139 27 131)(6 42 148 36 140 28 132)(7 43 149 37 141 29 133)(8 44 150 38 142 30 134)(9 21 61 165 53 157 122)(10 22 62 166 54 158 123)(11 23 63 167 55 159 124)(12 24 64 168 56 160 125)(13 17 57 161 49 153 126)(14 18 58 162 50 154 127)(15 19 59 163 51 155 128)(16 20 60 164 52 156 121)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 119 217 105 209 97 201)(90 120 218 106 210 98 202)(91 113 219 107 211 99 203)(92 114 220 108 212 100 204)(93 115 221 109 213 101 205)(94 116 222 110 214 102 206)(95 117 223 111 215 103 207)(96 118 224 112 216 104 208)
(1 175)(2 172)(3 169)(4 174)(5 171)(6 176)(7 173)(8 170)(9 219)(10 224)(11 221)(12 218)(13 223)(14 220)(15 217)(16 222)(17 111)(18 108)(19 105)(20 110)(21 107)(22 112)(23 109)(24 106)(25 177)(26 182)(27 179)(28 184)(29 181)(30 178)(31 183)(32 180)(33 185)(34 190)(35 187)(36 192)(37 189)(38 186)(39 191)(40 188)(41 195)(42 200)(43 197)(44 194)(45 199)(46 196)(47 193)(48 198)(49 207)(50 204)(51 201)(52 206)(53 203)(54 208)(55 205)(56 202)(57 215)(58 212)(59 209)(60 214)(61 211)(62 216)(63 213)(64 210)(65 129)(66 134)(67 131)(68 136)(69 133)(70 130)(71 135)(72 132)(73 137)(74 142)(75 139)(76 144)(77 141)(78 138)(79 143)(80 140)(81 145)(82 150)(83 147)(84 152)(85 149)(86 146)(87 151)(88 148)(89 155)(90 160)(91 157)(92 154)(93 159)(94 156)(95 153)(96 158)(97 163)(98 168)(99 165)(100 162)(101 167)(102 164)(103 161)(104 166)(113 122)(114 127)(115 124)(116 121)(117 126)(118 123)(119 128)(120 125)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
(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 91 175 157)(2 154 176 96)(3 89 169 155)(4 160 170 94)(5 95 171 153)(6 158 172 92)(7 93 173 159)(8 156 174 90)(9 151 219 87)(10 84 220 148)(11 149 221 85)(12 82 222 146)(13 147 223 83)(14 88 224 152)(15 145 217 81)(16 86 218 150)(17 35 111 187)(18 192 112 40)(19 33 105 185)(20 190 106 38)(21 39 107 191)(22 188 108 36)(23 37 109 189)(24 186 110 34)(25 97 177 163)(26 168 178 102)(27 103 179 161)(28 166 180 100)(29 101 181 167)(30 164 182 98)(31 99 183 165)(32 162 184 104)(41 117 195 126)(42 123 196 114)(43 115 197 124)(44 121 198 120)(45 113 199 122)(46 127 200 118)(47 119 193 128)(48 125 194 116)(49 131 207 67)(50 72 208 136)(51 129 201 65)(52 70 202 134)(53 135 203 71)(54 68 204 132)(55 133 205 69)(56 66 206 130)(57 139 215 75)(58 80 216 144)(59 137 209 73)(60 78 210 142)(61 143 211 79)(62 76 212 140)(63 141 213 77)(64 74 214 138)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AV 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + - + - image C1 C2 C2 C4 C4 C7 C14 C14 C28 C28 D4 Q8 C7×D4 C7×Q8 C4.D4 C4.10D4 C7×C4.D4 C7×C4.10D4 kernel C7×C22.C42 C14×C4⋊C4 C14×M4(2) C7×M4(2) C22×C28 C22.C42 C2×C4⋊C4 C2×M4(2) M4(2) C22×C4 C2×C28 C2×C28 C2×C4 C2×C4 C14 C14 C2 C2 # reps 1 1 2 8 4 6 6 12 48 24 3 1 18 6 1 1 6 6

Matrix representation of C7×C22.C42 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 49 0 0 0 0 0 0 49 0 0 0 0 0 0 49 0 0 0 0 0 0 49
,
 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 86 101 0 0 0 112 9 107 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 103 64 0 0 0 0 92 10 0 0 0 0 0 0 101 27 55 9 0 0 107 104 16 106 0 0 0 111 9 107 0 0 2 0 27 12
,
 104 2 0 0 0 0 72 9 0 0 0 0 0 0 112 26 12 90 0 0 26 1 90 45 0 0 0 0 1 87 0 0 0 0 87 112

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,49,0,0,0,0,0,0,49,0,0,0,0,0,0,49,0,0,0,0,0,0,49],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,86,9,1,0,0,0,101,107,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[103,92,0,0,0,0,64,10,0,0,0,0,0,0,101,107,0,2,0,0,27,104,111,0,0,0,55,16,9,27,0,0,9,106,107,12],[104,72,0,0,0,0,2,9,0,0,0,0,0,0,112,26,0,0,0,0,26,1,0,0,0,0,12,90,1,87,0,0,90,45,87,112] >;

C7×C22.C42 in GAP, Magma, Sage, TeX

C_7\times C_2^2.C_4^2
% in TeX

G:=Group("C7xC2^2.C4^2");
// GroupNames label

G:=SmallGroup(448,147);
// by ID

G=gap.SmallGroup(448,147);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,7059,4911,172]);
// Polycyclic

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

׿
×
𝔽