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G = D7×C22×C8  order 448 = 26·7

Direct product of C22×C8 and D7

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×C22×C8, C56⋊11C23, C28.65C24, C7⋊1(C23×C8), C7⋊C8⋊14C23, C14⋊1(C22×C8), (C2×C56)⋊47C22, (C22×C56)⋊17C2, C23.65(C4×D7), C4.64(C23×D7), C14.28(C23×C4), (C4×D7).39C23, (C23×D7).10C4, C28.144(C22×C4), (C2×C28).878C23, D14.25(C22×C4), (C22×C4).469D14, Dic7.26(C22×C4), (C22×Dic7).20C4, (C22×C28).566C22, (C2×C14)⋊6(C2×C8), (C2×C4×D7).25C4, C4.119(C2×C4×D7), C2.2(D7×C22×C4), (C2×C7⋊C8)⋊49C22, (C22×C7⋊C8)⋊24C2, C22.74(C2×C4×D7), (C4×D7).39(C2×C4), (C2×C4).186(C4×D7), (D7×C22×C4).25C2, (C2×C28).256(C2×C4), (C2×C4×D7).309C22, (C22×D7).75(C2×C4), (C2×C4).822(C22×D7), (C22×C14).101(C2×C4), (C2×C14).154(C22×C4), (C2×Dic7).112(C2×C4), SmallGroup(448,1189)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — D7×C22×C8
Chief series
C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — D7×C22×C4 — D7×C22×C8
Lower central
C7 — D7×C22×C8
Upper central
C1 — C22×C8

Generators and relations for D7×C22×C8
 G = < a,b,c,d,e | a2=b2=c8=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1124 in 338 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C23, D7, C14, C14, C2×C8, C2×C8, C22×C4, C22×C4, C24, Dic7, C28, C28, D14, C2×C14, C22×C8, C22×C8, C23×C4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×C14, C23×C8, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, D7×C2×C8, C22×C7⋊C8, C22×C56, D7×C22×C4, D7×C22×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, C24, D14, C22×C8, C23×C4, C4×D7, C22×D7, C23×C8, C8×D7, C2×C4×D7, C23×D7, D7×C2×C8, D7×C22×C4, D7×C22×C8

Smallest permutation representation of D7×C22×C8
►On 224 points
Generators in S224
(1 168)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 119)(26 120)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 127)(34 128)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 135)(42 136)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 143)(50 144)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 173)(58 174)(59 175)(60 176)(61 169)(62 170)(63 171)(64 172)(65 181)(66 182)(67 183)(68 184)(69 177)(70 178)(71 179)(72 180)(73 189)(74 190)(75 191)(76 192)(77 185)(78 186)(79 187)(80 188)(81 197)(82 198)(83 199)(84 200)(85 193)(86 194)(87 195)(88 196)(89 205)(90 206)(91 207)(92 208)(93 201)(94 202)(95 203)(96 204)(97 213)(98 214)(99 215)(100 216)(101 209)(102 210)(103 211)(104 212)(105 221)(106 222)(107 223)(108 224)(109 217)(110 218)(111 219)(112 220)

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

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

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

160 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4P7A7B7C8A···8P8Q···8AF14A···14U28A···28X56A···56AV
order12···22···24···44···47778···88···814···1428···2856···56
size11···17···71···17···72221···17···72···22···22···2

160 irreducible representations

dim111111111222222
type++++++++
imageC1C2C2C2C2C4C4C4C8D7D14D14C4×D7C4×D7C8×D7
kernelD7×C22×C8D7×C2×C8C22×C7⋊C8C22×C56D7×C22×C4C2×C4×D7C22×Dic7C23×D7C22×D7C22×C8C2×C8C22×C4C2×C4C23C22
# reps112111122232318318648

Matrix representation of D7×C22×C8 ►in GL4(𝔽113) generated by

112000
011200
001120
000112
,
1000
011200
001120
000112
,
44000
09800
00440
00044
,
1000
0100
00882
00112104
,
112000
0100
008859
0011225
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[44,0,0,0,0,98,0,0,0,0,44,0,0,0,0,44],[1,0,0,0,0,1,0,0,0,0,88,112,0,0,2,104],[112,0,0,0,0,1,0,0,0,0,88,112,0,0,59,25] >;
 

D7×C22×C8 in GAP, Magma, Sage, TeX 

D_7\times C_2^2\times C_8
 
% in TeX
 
G:=Group("D7xC2^2xC8");
 
// GroupNames label
 
G:=SmallGroup(448,1189);
 
// by ID
 
G=gap.SmallGroup(448,1189);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,80,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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