metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.47D28, M4(2)⋊1Dic7, C56⋊3(C2×C4), C8⋊Dic7⋊3C2, C8⋊1(C2×Dic7), C56⋊1C4⋊17C2, (C2×C8).75D14, C28.21(C4⋊C4), C28.76(C2×Q8), (C2×C28).26Q8, (C2×C4).149D28, (C2×C28).167D4, (C7×M4(2))⋊1C4, C4.6(C4⋊Dic7), C2.3(C8⋊D14), (C2×C56).61C22, C4.42(C2×Dic14), (C2×C4).15Dic14, C22.56(C2×D28), (C2×M4(2)).1D7, C14.19(C8⋊C22), C7⋊5(M4(2)⋊C4), C28.173(C22×C4), (C2×C28).772C23, C2.4(C8.D14), (C22×C14).100D4, (C22×C4).133D14, (C14×M4(2)).1C2, C22.6(C4⋊Dic7), C4.27(C22×Dic7), C14.20(C8.C22), C4⋊Dic7.284C22, (C22×C28).180C22, C23.21D14.17C2, C14.50(C2×C4⋊C4), C2.14(C2×C4⋊Dic7), (C2×C14).16(C4⋊C4), (C2×C28).100(C2×C4), (C2×C14).162(C2×D4), (C2×C4⋊Dic7).39C2, (C2×C4).21(C2×Dic7), (C2×C4).720(C22×D7), SmallGroup(448,655)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.47D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d27 >
Subgroups: 484 in 118 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C56, C2×Dic7, C2×C28, C2×C28, C22×C14, M4(2)⋊C4, C4×Dic7, C4⋊Dic7, C4⋊Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×Dic7, C22×C28, C8⋊Dic7, C56⋊1C4, C2×C4⋊Dic7, C23.21D14, C14×M4(2), C23.47D28
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, C8⋊C22, C8.C22, Dic14, D28, C2×Dic7, C22×D7, M4(2)⋊C4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, C8⋊D14, C8.D14, C2×C4⋊Dic7, C23.47D28
►On 224 points
Generators in S224
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 42)(16 44)(18 46)(20 48)(22 50)(24 52)(26 54)(28 56)(57 85)(59 87)(61 89)(63 91)(65 93)(67 95)(69 97)(71 99)(73 101)(75 103)(77 105)(79 107)(81 109)(83 111)(114 142)(116 144)(118 146)(120 148)(122 150)(124 152)(126 154)(128 156)(130 158)(132 160)(134 162)(136 164)(138 166)(140 168)(170 198)(172 200)(174 202)(176 204)(178 206)(180 208)(182 210)(184 212)(186 214)(188 216)(190 218)(192 220)(194 222)(196 224)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(113 217)(114 218)(115 219)(116 220)(117 221)(118 222)(119 223)(120 224)(121 169)(122 170)(123 171)(124 172)(125 173)(126 174)(127 175)(128 176)(129 177)(130 178)(131 179)(132 180)(133 181)(134 182)(135 183)(136 184)(137 185)(138 186)(139 187)(140 188)(141 189)(142 190)(143 191)(144 192)(145 193)(146 194)(147 195)(148 196)(149 197)(150 198)(151 199)(152 200)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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)
(1 116 108 192)(2 143 109 219)(3 114 110 190)(4 141 111 217)(5 168 112 188)(6 139 57 215)(7 166 58 186)(8 137 59 213)(9 164 60 184)(10 135 61 211)(11 162 62 182)(12 133 63 209)(13 160 64 180)(14 131 65 207)(15 158 66 178)(16 129 67 205)(17 156 68 176)(18 127 69 203)(19 154 70 174)(20 125 71 201)(21 152 72 172)(22 123 73 199)(23 150 74 170)(24 121 75 197)(25 148 76 224)(26 119 77 195)(27 146 78 222)(28 117 79 193)(29 144 80 220)(30 115 81 191)(31 142 82 218)(32 113 83 189)(33 140 84 216)(34 167 85 187)(35 138 86 214)(36 165 87 185)(37 136 88 212)(38 163 89 183)(39 134 90 210)(40 161 91 181)(41 132 92 208)(42 159 93 179)(43 130 94 206)(44 157 95 177)(45 128 96 204)(46 155 97 175)(47 126 98 202)(48 153 99 173)(49 124 100 200)(50 151 101 171)(51 122 102 198)(52 149 103 169)(53 120 104 196)(54 147 105 223)(55 118 106 194)(56 145 107 221)
G:=sub<Sym(224)| (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111)(114,142)(116,144)(118,146)(120,148)(122,150)(124,152)(126,154)(128,156)(130,158)(132,160)(134,162)(136,164)(138,166)(140,168)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,116,108,192)(2,143,109,219)(3,114,110,190)(4,141,111,217)(5,168,112,188)(6,139,57,215)(7,166,58,186)(8,137,59,213)(9,164,60,184)(10,135,61,211)(11,162,62,182)(12,133,63,209)(13,160,64,180)(14,131,65,207)(15,158,66,178)(16,129,67,205)(17,156,68,176)(18,127,69,203)(19,154,70,174)(20,125,71,201)(21,152,72,172)(22,123,73,199)(23,150,74,170)(24,121,75,197)(25,148,76,224)(26,119,77,195)(27,146,78,222)(28,117,79,193)(29,144,80,220)(30,115,81,191)(31,142,82,218)(32,113,83,189)(33,140,84,216)(34,167,85,187)(35,138,86,214)(36,165,87,185)(37,136,88,212)(38,163,89,183)(39,134,90,210)(40,161,91,181)(41,132,92,208)(42,159,93,179)(43,130,94,206)(44,157,95,177)(45,128,96,204)(46,155,97,175)(47,126,98,202)(48,153,99,173)(49,124,100,200)(50,151,101,171)(51,122,102,198)(52,149,103,169)(53,120,104,196)(54,147,105,223)(55,118,106,194)(56,145,107,221)>;
G:=Group( (2,30)(4,32)(6,34)(8,36)(10,38)(12,40)(14,42)(16,44)(18,46)(20,48)(22,50)(24,52)(26,54)(28,56)(57,85)(59,87)(61,89)(63,91)(65,93)(67,95)(69,97)(71,99)(73,101)(75,103)(77,105)(79,107)(81,109)(83,111)(114,142)(116,144)(118,146)(120,148)(122,150)(124,152)(126,154)(128,156)(130,158)(132,160)(134,162)(136,164)(138,166)(140,168)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192)(145,193)(146,194)(147,195)(148,196)(149,197)(150,198)(151,199)(152,200)(153,201)(154,202)(155,203)(156,204)(157,205)(158,206)(159,207)(160,208)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,116,108,192)(2,143,109,219)(3,114,110,190)(4,141,111,217)(5,168,112,188)(6,139,57,215)(7,166,58,186)(8,137,59,213)(9,164,60,184)(10,135,61,211)(11,162,62,182)(12,133,63,209)(13,160,64,180)(14,131,65,207)(15,158,66,178)(16,129,67,205)(17,156,68,176)(18,127,69,203)(19,154,70,174)(20,125,71,201)(21,152,72,172)(22,123,73,199)(23,150,74,170)(24,121,75,197)(25,148,76,224)(26,119,77,195)(27,146,78,222)(28,117,79,193)(29,144,80,220)(30,115,81,191)(31,142,82,218)(32,113,83,189)(33,140,84,216)(34,167,85,187)(35,138,86,214)(36,165,87,185)(37,136,88,212)(38,163,89,183)(39,134,90,210)(40,161,91,181)(41,132,92,208)(42,159,93,179)(43,130,94,206)(44,157,95,177)(45,128,96,204)(46,155,97,175)(47,126,98,202)(48,153,99,173)(49,124,100,200)(50,151,101,171)(51,122,102,198)(52,149,103,169)(53,120,104,196)(54,147,105,223)(55,118,106,194)(56,145,107,221) );
G=PermutationGroup([[(2,30),(4,32),(6,34),(8,36),(10,38),(12,40),(14,42),(16,44),(18,46),(20,48),(22,50),(24,52),(26,54),(28,56),(57,85),(59,87),(61,89),(63,91),(65,93),(67,95),(69,97),(71,99),(73,101),(75,103),(77,105),(79,107),(81,109),(83,111),(114,142),(116,144),(118,146),(120,148),(122,150),(124,152),(126,154),(128,156),(130,158),(132,160),(134,162),(136,164),(138,166),(140,168),(170,198),(172,200),(174,202),(176,204),(178,206),(180,208),(182,210),(184,212),(186,214),(188,216),(190,218),(192,220),(194,222),(196,224)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(113,217),(114,218),(115,219),(116,220),(117,221),(118,222),(119,223),(120,224),(121,169),(122,170),(123,171),(124,172),(125,173),(126,174),(127,175),(128,176),(129,177),(130,178),(131,179),(132,180),(133,181),(134,182),(135,183),(136,184),(137,185),(138,186),(139,187),(140,188),(141,189),(142,190),(143,191),(144,192),(145,193),(146,194),(147,195),(148,196),(149,197),(150,198),(151,199),(152,200),(153,201),(154,202),(155,203),(156,204),(157,205),(158,206),(159,207),(160,208),(161,209),(162,210),(163,211),(164,212),(165,213),(166,214),(167,215),(168,216)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(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)], [(1,116,108,192),(2,143,109,219),(3,114,110,190),(4,141,111,217),(5,168,112,188),(6,139,57,215),(7,166,58,186),(8,137,59,213),(9,164,60,184),(10,135,61,211),(11,162,62,182),(12,133,63,209),(13,160,64,180),(14,131,65,207),(15,158,66,178),(16,129,67,205),(17,156,68,176),(18,127,69,203),(19,154,70,174),(20,125,71,201),(21,152,72,172),(22,123,73,199),(23,150,74,170),(24,121,75,197),(25,148,76,224),(26,119,77,195),(27,146,78,222),(28,117,79,193),(29,144,80,220),(30,115,81,191),(31,142,82,218),(32,113,83,189),(33,140,84,216),(34,167,85,187),(35,138,86,214),(36,165,87,185),(37,136,88,212),(38,163,89,183),(39,134,90,210),(40,161,91,181),(41,132,92,208),(42,159,93,179),(43,130,94,206),(44,157,95,177),(45,128,96,204),(46,155,97,175),(47,126,98,202),(48,153,99,173),(49,124,100,200),(50,151,101,171),(51,122,102,198),(52,149,103,169),(53,120,104,196),(54,147,105,223),(55,118,106,194),(56,145,107,221)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | D14 | Dic7 | D14 | Dic14 | D28 | D28 | C8⋊C22 | C8.C22 | C8⋊D14 | C8.D14 |
| kernel | C23.47D28 | C8⋊Dic7 | C56⋊1C4 | C2×C4⋊Dic7 | C23.21D14 | C14×M4(2) | C7×M4(2) | C2×C28 | C2×C28 | C22×C14 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C23 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 6 | 12 | 3 | 12 | 6 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C23.47D28 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 41 | 112 | 0 |
| 0 | 0 | 33 | 41 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 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 |
| 13 | 77 | 0 | 0 | 0 | 0 |
| 32 | 94 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 89 | 14 | 23 |
| 0 | 0 | 35 | 88 | 104 | 24 |
| 0 | 0 | 85 | 75 | 103 | 24 |
| 0 | 0 | 98 | 39 | 103 | 24 |
| 112 | 83 | 0 | 0 | 0 | 0 |
| 98 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 51 | 89 | 16 |
| 0 | 0 | 49 | 78 | 94 | 40 |
| 0 | 0 | 104 | 22 | 62 | 67 |
| 0 | 0 | 69 | 84 | 62 | 67 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,33,33,0,0,0,1,41,41,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[13,32,0,0,0,0,77,94,0,0,0,0,0,0,11,35,85,98,0,0,89,88,75,39,0,0,14,104,103,103,0,0,23,24,24,24],[112,98,0,0,0,0,83,1,0,0,0,0,0,0,19,49,104,69,0,0,51,78,22,84,0,0,89,94,62,62,0,0,16,40,67,67] >;
C23.47D28 in GAP, Magma, Sage, TeX
C_2^3._{47}D_{28} % in TeX
G:=Group("C2^3.47D28"); // GroupNames label
G:=SmallGroup(448,655);
// by ID
G=gap.SmallGroup(448,655);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,387,100,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^27>;
// generators/relations