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G = C23×D28  order 448 = 26·7

Direct product of C23 and D28

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

Aliases: C23×D28, C28⋊2C24, D14⋊1C24, C14.3C25, C24.82D14, C7⋊1(D4×C23), (C23×C4)⋊7D7, C4⋊2(C23×D7), (C23×C28)⋊9C2, (D7×C24)⋊4C2, C14⋊1(C22×D4), C2.4(D7×C24), (C2×C28)⋊14C23, (C22×C14)⋊16D4, (C22×C4)⋊45D14, (C22×D7)⋊7C23, (C2×C14).325C24, (C22×C28)⋊61C22, (C23×D7)⋊22C22, C22.53(C23×D7), C23.346(C22×D7), (C22×C14).432C23, (C23×C14).115C22, (C2×C14)⋊12(C2×D4), (C2×C4)⋊11(C22×D7), SmallGroup(448,1367)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — C23×D28
Chief series
C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C24 — C23×D28
Lower central
C7 — C14 — C23×D28

Subgroups: 6788 in 1362 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×16], C4 [×8], C22 [×35], C22 [×128], C7, C2×C4 [×28], D4 [×64], C23 [×15], C23 [×168], D7 [×16], C14, C14 [×14], C22×C4 [×14], C2×D4 [×112], C24, C24 [×44], C28 [×8], D14 [×16], D14 [×112], C2×C14 [×35], C23×C4, C22×D4 [×28], C25 [×2], D28 [×64], C2×C28 [×28], C22×D7 [×56], C22×D7 [×112], C22×C14 [×15], D4×C23, C2×D28 [×112], C22×C28 [×14], C23×D7 [×28], C23×D7 [×16], C23×C14, C22×D28 [×28], C23×C28, D7×C24 [×2], C23×D28

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], D7, C2×D4 [×28], C24 [×31], D14 [×15], C22×D4 [×14], C25, D28 [×8], C22×D7 [×35], D4×C23, C2×D28 [×28], C23×D7 [×15], C22×D28 [×14], D7×C24, C23×D28

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d28=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
(1 184)(2 185)(3 186)(4 187)(5 188)(6 189)(7 190)(8 191)(9 192)(10 193)(11 194)(12 195)(13 196)(14 169)(15 170)(16 171)(17 172)(18 173)(19 174)(20 175)(21 176)(22 177)(23 178)(24 179)(25 180)(26 181)(27 182)(28 183)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 85)(41 86)(42 87)(43 88)(44 89)(45 90)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 100)(56 101)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 211)(70 212)(71 213)(72 214)(73 215)(74 216)(75 217)(76 218)(77 219)(78 220)(79 221)(80 222)(81 223)(82 224)(83 197)(84 198)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 167)(122 168)(123 141)(124 142)(125 143)(126 144)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽29)

280000
01000
002800
00010
00001
,
280000
028000
002800
000280
000028
,
10000
01000
002800
00010
00001
,
10000
01000
00100
000422
000249
,
10000
028000
00100
000025
00070

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,24,0,0,0,22,9],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,25,0] >;
 

136 conjugacy classes

class 1 2A···2O2P···2AE4A···4H7A7B7C14A···14AS28A···28AV
order12···22···24···477714···1428···28
size11···114···142···22222···22···2

136 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D7D14D14D28
kernelC23×D28C22×D28C23×C28D7×C24C22×C14C23×C4C22×C4C24C23
# reps128128342348

In GAP, Magma, Sage, TeX 

C_2^3\times D_{28}
 
% in TeX
 
G:=Group("C2^3xD28");
 
// GroupNames label
 
G:=SmallGroup(448,1367);
 
// by ID
 
G=gap.SmallGroup(448,1367);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1684,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=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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