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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, C4.64(C23×D7), C23.65(C4×D7), C14.28(C23×C4), (C4×D7).39C23, (C23×D7).10C4, (C2×C28).878C23, C28.144(C22×C4), 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), (C22×C7⋊C8)⋊24C2, (C2×C7⋊C8)⋊49C22, 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), (C2×C14).154(C22×C4), (C22×C14).101(C2×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

Subgroups: 1124 in 338 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×28], C7, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], D7 [×8], C14, C14 [×6], C2×C8 [×6], C2×C8 [×22], C22×C4, C22×C4 [×13], C24, Dic7 [×4], C28, C28 [×3], D14 [×28], C2×C14 [×7], C22×C8, C22×C8 [×13], C23×C4, C7⋊C8 [×4], C56 [×4], C4×D7 [×16], C2×Dic7 [×6], C2×C28 [×6], C22×D7 [×14], C22×C14, C23×C8, C8×D7 [×16], C2×C7⋊C8 [×6], C2×C56 [×6], C2×C4×D7 [×12], C22×Dic7, C22×C28, C23×D7, D7×C2×C8 [×12], C22×C7⋊C8, C22×C56, D7×C22×C4, D7×C22×C8

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], D7, C2×C8 [×28], C22×C4 [×14], C24, D14 [×7], C22×C8 [×14], C23×C4, C4×D7 [×4], C22×D7 [×7], C23×C8, C8×D7 [×4], C2×C4×D7 [×6], C23×D7, D7×C2×C8 [×6], D7×C22×C4, D7×C22×C8

Generators and relations
 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 >

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
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