metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊2D28, C24.15D14, (C2×C28)⋊6D4, (C2×Dic7)⋊5D4, (C22×D7)⋊4D4, (C22×C14)⋊7D4, (C22×D28)⋊3C2, C7⋊2(C23⋊2D4), C14.33C22≀C2, C2.7(C28⋊7D4), C2.6(C28⋊D4), (C22×C4).35D14, C22.242(D4×D7), C2.8(C23⋊D14), C14.59(C4⋊D4), C14.13(C4⋊1D4), C22.126(C2×D28), C2.34(C22⋊D28), C2.34(D14⋊D4), C14.C42⋊32C2, (C23×C14).43C22, (C22×C28).61C22, (C23×D7).16C22, C23.372(C22×D7), C22.100(C4○D28), (C22×C14).334C23, (C22×Dic7).46C22, (C2×D14⋊C4)⋊8C2, (C2×C4)⋊3(C7⋊D4), (C2×C22⋊C4)⋊8D7, (C22×C7⋊D4)⋊1C2, (C14×C22⋊C4)⋊11C2, (C2×C14).325(C2×D4), (C2×C14).80(C4○D4), C22.128(C2×C7⋊D4), SmallGroup(448,494)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊2D28
G = < a,b,c,d,e | a2=b2=c2=d28=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1972 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C22×D4, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23⋊2D4, D14⋊C4, C7×C22⋊C4, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C14×C22⋊C4, C22×D28, C22×C7⋊D4, C23⋊2D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22≀C2, C4⋊D4, C4⋊1D4, D28, C7⋊D4, C22×D7, C23⋊2D4, C2×D28, C4○D28, D4×D7, C2×C7⋊D4, C22⋊D28, D14⋊D4, C28⋊7D4, C23⋊D14, C28⋊D4, C23⋊2D28
►On 224 points
Generators in S224
(2 88)(4 90)(6 92)(8 94)(10 96)(12 98)(14 100)(16 102)(18 104)(20 106)(22 108)(24 110)(26 112)(28 86)(29 58)(31 60)(33 62)(35 64)(37 66)(39 68)(41 70)(43 72)(45 74)(47 76)(49 78)(51 80)(53 82)(55 84)(113 141)(114 171)(115 143)(116 173)(117 145)(118 175)(119 147)(120 177)(121 149)(122 179)(123 151)(124 181)(125 153)(126 183)(127 155)(128 185)(129 157)(130 187)(131 159)(132 189)(133 161)(134 191)(135 163)(136 193)(137 165)(138 195)(139 167)(140 169)(142 219)(144 221)(146 223)(148 197)(150 199)(152 201)(154 203)(156 205)(158 207)(160 209)(162 211)(164 213)(166 215)(168 217)(170 218)(172 220)(174 222)(176 224)(178 198)(180 200)(182 202)(184 204)(186 206)(188 208)(190 210)(192 212)(194 214)(196 216)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 88)(30 89)(31 90)(32 91)(33 92)(34 93)(35 94)(36 95)(37 96)(38 97)(39 98)(40 99)(41 100)(42 101)(43 102)(44 103)(45 104)(46 105)(47 106)(48 107)(49 108)(50 109)(51 110)(52 111)(53 112)(54 85)(55 86)(56 87)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 169)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 209)(161 210)(162 211)(163 212)(164 213)(165 214)(166 215)(167 216)(168 217)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 85)(28 86)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 57)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 197)(121 198)(122 199)(123 200)(124 201)(125 202)(126 203)(127 204)(128 205)(129 206)(130 207)(131 208)(132 209)(133 210)(134 211)(135 212)(136 213)(137 214)(138 215)(139 216)(140 217)(141 170)(142 171)(143 172)(144 173)(145 174)(146 175)(147 176)(148 177)(149 178)(150 179)(151 180)(152 181)(153 182)(154 183)(155 184)(156 185)(157 186)(158 187)(159 188)(160 189)(161 190)(162 191)(163 192)(164 193)(165 194)(166 195)(167 196)(168 169)
(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)
(1 145)(2 144)(3 143)(4 142)(5 141)(6 168)(7 167)(8 166)(9 165)(10 164)(11 163)(12 162)(13 161)(14 160)(15 159)(16 158)(17 157)(18 156)(19 155)(20 154)(21 153)(22 152)(23 151)(24 150)(25 149)(26 148)(27 147)(28 146)(29 116)(30 115)(31 114)(32 113)(33 140)(34 139)(35 138)(36 137)(37 136)(38 135)(39 134)(40 133)(41 132)(42 131)(43 130)(44 129)(45 128)(46 127)(47 126)(48 125)(49 124)(50 123)(51 122)(52 121)(53 120)(54 119)(55 118)(56 117)(57 222)(58 221)(59 220)(60 219)(61 218)(62 217)(63 216)(64 215)(65 214)(66 213)(67 212)(68 211)(69 210)(70 209)(71 208)(72 207)(73 206)(74 205)(75 204)(76 203)(77 202)(78 201)(79 200)(80 199)(81 198)(82 197)(83 224)(84 223)(85 176)(86 175)(87 174)(88 173)(89 172)(90 171)(91 170)(92 169)(93 196)(94 195)(95 194)(96 193)(97 192)(98 191)(99 190)(100 189)(101 188)(102 187)(103 186)(104 185)(105 184)(106 183)(107 182)(108 181)(109 180)(110 179)(111 178)(112 177)
G:=sub<Sym(224)| (2,88)(4,90)(6,92)(8,94)(10,96)(12,98)(14,100)(16,102)(18,104)(20,106)(22,108)(24,110)(26,112)(28,86)(29,58)(31,60)(33,62)(35,64)(37,66)(39,68)(41,70)(43,72)(45,74)(47,76)(49,78)(51,80)(53,82)(55,84)(113,141)(114,171)(115,143)(116,173)(117,145)(118,175)(119,147)(120,177)(121,149)(122,179)(123,151)(124,181)(125,153)(126,183)(127,155)(128,185)(129,157)(130,187)(131,159)(132,189)(133,161)(134,191)(135,163)(136,193)(137,165)(138,195)(139,167)(140,169)(142,219)(144,221)(146,223)(148,197)(150,199)(152,201)(154,203)(156,205)(158,207)(160,209)(162,211)(164,213)(166,215)(168,217)(170,218)(172,220)(174,222)(176,224)(178,198)(180,200)(182,202)(184,204)(186,206)(188,208)(190,210)(192,212)(194,214)(196,216), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,85)(55,86)(56,87)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,169)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,57)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,169), (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), (1,145)(2,144)(3,143)(4,142)(5,141)(6,168)(7,167)(8,166)(9,165)(10,164)(11,163)(12,162)(13,161)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,116)(30,115)(31,114)(32,113)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,222)(58,221)(59,220)(60,219)(61,218)(62,217)(63,216)(64,215)(65,214)(66,213)(67,212)(68,211)(69,210)(70,209)(71,208)(72,207)(73,206)(74,205)(75,204)(76,203)(77,202)(78,201)(79,200)(80,199)(81,198)(82,197)(83,224)(84,223)(85,176)(86,175)(87,174)(88,173)(89,172)(90,171)(91,170)(92,169)(93,196)(94,195)(95,194)(96,193)(97,192)(98,191)(99,190)(100,189)(101,188)(102,187)(103,186)(104,185)(105,184)(106,183)(107,182)(108,181)(109,180)(110,179)(111,178)(112,177)>;
G:=Group( (2,88)(4,90)(6,92)(8,94)(10,96)(12,98)(14,100)(16,102)(18,104)(20,106)(22,108)(24,110)(26,112)(28,86)(29,58)(31,60)(33,62)(35,64)(37,66)(39,68)(41,70)(43,72)(45,74)(47,76)(49,78)(51,80)(53,82)(55,84)(113,141)(114,171)(115,143)(116,173)(117,145)(118,175)(119,147)(120,177)(121,149)(122,179)(123,151)(124,181)(125,153)(126,183)(127,155)(128,185)(129,157)(130,187)(131,159)(132,189)(133,161)(134,191)(135,163)(136,193)(137,165)(138,195)(139,167)(140,169)(142,219)(144,221)(146,223)(148,197)(150,199)(152,201)(154,203)(156,205)(158,207)(160,209)(162,211)(164,213)(166,215)(168,217)(170,218)(172,220)(174,222)(176,224)(178,198)(180,200)(182,202)(184,204)(186,206)(188,208)(190,210)(192,212)(194,214)(196,216), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,88)(30,89)(31,90)(32,91)(33,92)(34,93)(35,94)(36,95)(37,96)(38,97)(39,98)(40,99)(41,100)(42,101)(43,102)(44,103)(45,104)(46,105)(47,106)(48,107)(49,108)(50,109)(51,110)(52,111)(53,112)(54,85)(55,86)(56,87)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,169)(141,218)(142,219)(143,220)(144,221)(145,222)(146,223)(147,224)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,209)(161,210)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,58)(30,59)(31,60)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,80)(52,81)(53,82)(54,83)(55,84)(56,57)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,197)(121,198)(122,199)(123,200)(124,201)(125,202)(126,203)(127,204)(128,205)(129,206)(130,207)(131,208)(132,209)(133,210)(134,211)(135,212)(136,213)(137,214)(138,215)(139,216)(140,217)(141,170)(142,171)(143,172)(144,173)(145,174)(146,175)(147,176)(148,177)(149,178)(150,179)(151,180)(152,181)(153,182)(154,183)(155,184)(156,185)(157,186)(158,187)(159,188)(160,189)(161,190)(162,191)(163,192)(164,193)(165,194)(166,195)(167,196)(168,169), (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), (1,145)(2,144)(3,143)(4,142)(5,141)(6,168)(7,167)(8,166)(9,165)(10,164)(11,163)(12,162)(13,161)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,153)(22,152)(23,151)(24,150)(25,149)(26,148)(27,147)(28,146)(29,116)(30,115)(31,114)(32,113)(33,140)(34,139)(35,138)(36,137)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,222)(58,221)(59,220)(60,219)(61,218)(62,217)(63,216)(64,215)(65,214)(66,213)(67,212)(68,211)(69,210)(70,209)(71,208)(72,207)(73,206)(74,205)(75,204)(76,203)(77,202)(78,201)(79,200)(80,199)(81,198)(82,197)(83,224)(84,223)(85,176)(86,175)(87,174)(88,173)(89,172)(90,171)(91,170)(92,169)(93,196)(94,195)(95,194)(96,193)(97,192)(98,191)(99,190)(100,189)(101,188)(102,187)(103,186)(104,185)(105,184)(106,183)(107,182)(108,181)(109,180)(110,179)(111,178)(112,177) );
G=PermutationGroup([[(2,88),(4,90),(6,92),(8,94),(10,96),(12,98),(14,100),(16,102),(18,104),(20,106),(22,108),(24,110),(26,112),(28,86),(29,58),(31,60),(33,62),(35,64),(37,66),(39,68),(41,70),(43,72),(45,74),(47,76),(49,78),(51,80),(53,82),(55,84),(113,141),(114,171),(115,143),(116,173),(117,145),(118,175),(119,147),(120,177),(121,149),(122,179),(123,151),(124,181),(125,153),(126,183),(127,155),(128,185),(129,157),(130,187),(131,159),(132,189),(133,161),(134,191),(135,163),(136,193),(137,165),(138,195),(139,167),(140,169),(142,219),(144,221),(146,223),(148,197),(150,199),(152,201),(154,203),(156,205),(158,207),(160,209),(162,211),(164,213),(166,215),(168,217),(170,218),(172,220),(174,222),(176,224),(178,198),(180,200),(182,202),(184,204),(186,206),(188,208),(190,210),(192,212),(194,214),(196,216)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,88),(30,89),(31,90),(32,91),(33,92),(34,93),(35,94),(36,95),(37,96),(38,97),(39,98),(40,99),(41,100),(42,101),(43,102),(44,103),(45,104),(46,105),(47,106),(48,107),(49,108),(50,109),(51,110),(52,111),(53,112),(54,85),(55,86),(56,87),(113,170),(114,171),(115,172),(116,173),(117,174),(118,175),(119,176),(120,177),(121,178),(122,179),(123,180),(124,181),(125,182),(126,183),(127,184),(128,185),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192),(136,193),(137,194),(138,195),(139,196),(140,169),(141,218),(142,219),(143,220),(144,221),(145,222),(146,223),(147,224),(148,197),(149,198),(150,199),(151,200),(152,201),(153,202),(154,203),(155,204),(156,205),(157,206),(158,207),(159,208),(160,209),(161,210),(162,211),(163,212),(164,213),(165,214),(166,215),(167,216),(168,217)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,85),(28,86),(29,58),(30,59),(31,60),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,80),(52,81),(53,82),(54,83),(55,84),(56,57),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,197),(121,198),(122,199),(123,200),(124,201),(125,202),(126,203),(127,204),(128,205),(129,206),(130,207),(131,208),(132,209),(133,210),(134,211),(135,212),(136,213),(137,214),(138,215),(139,216),(140,217),(141,170),(142,171),(143,172),(144,173),(145,174),(146,175),(147,176),(148,177),(149,178),(150,179),(151,180),(152,181),(153,182),(154,183),(155,184),(156,185),(157,186),(158,187),(159,188),(160,189),(161,190),(162,191),(163,192),(164,193),(165,194),(166,195),(167,196),(168,169)], [(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)], [(1,145),(2,144),(3,143),(4,142),(5,141),(6,168),(7,167),(8,166),(9,165),(10,164),(11,163),(12,162),(13,161),(14,160),(15,159),(16,158),(17,157),(18,156),(19,155),(20,154),(21,153),(22,152),(23,151),(24,150),(25,149),(26,148),(27,147),(28,146),(29,116),(30,115),(31,114),(32,113),(33,140),(34,139),(35,138),(36,137),(37,136),(38,135),(39,134),(40,133),(41,132),(42,131),(43,130),(44,129),(45,128),(46,127),(47,126),(48,125),(49,124),(50,123),(51,122),(52,121),(53,120),(54,119),(55,118),(56,117),(57,222),(58,221),(59,220),(60,219),(61,218),(62,217),(63,216),(64,215),(65,214),(66,213),(67,212),(68,211),(69,210),(70,209),(71,208),(72,207),(73,206),(74,205),(75,204),(76,203),(77,202),(78,201),(79,200),(80,199),(81,198),(82,197),(83,224),(84,223),(85,176),(86,175),(87,174),(88,173),(89,172),(90,171),(91,170),(92,169),(93,196),(94,195),(95,194),(96,193),(97,192),(98,191),(99,190),(100,189),(101,188),(102,187),(103,186),(104,185),(105,184),(106,183),(107,182),(108,181),(109,180),(110,179),(111,178),(112,177)]])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 28 | 28 | 28 | 28 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | C4○D28 | D4×D7 |
| kernel | C23⋊2D28 | C14.C42 | C2×D14⋊C4 | C14×C22⋊C4 | C22×D28 | C22×C7⋊D4 | C2×Dic7 | C2×C28 | C22×D7 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 3 | 2 | 6 | 3 | 12 | 12 | 12 | 12 |
Matrix representation of C23⋊2D28 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 23 | 11 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 25 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 2 |
| 0 | 0 | 0 | 0 | 7 | 4 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,0,0,0,0,28,0,0,0,0,0,0,1,4,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,27,28,0,0,0,0,0,0,8,23,0,0,0,0,0,11],[0,25,0,0,0,0,7,0,0,0,0,0,0,0,28,0,0,0,0,0,2,1,0,0,0,0,0,0,25,7,0,0,0,0,2,4] >;
C23⋊2D28 in GAP, Magma, Sage, TeX
C_2^3\rtimes_2D_{28} % in TeX
G:=Group("C2^3:2D28"); // GroupNames label
G:=SmallGroup(448,494);
// by ID
G=gap.SmallGroup(448,494);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,387,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=e^2=1,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e=a*b*c,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