metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.23D28, C4○D28⋊2C4, (C22×C8)⋊5D7, (C22×C56)⋊4C2, D28.23(C2×C4), (C2×C4).172D28, (C2×C8).295D14, (C2×C28).403D4, C28.411(C2×D4), C2.D56⋊44C2, C4.55(D14⋊C4), C14.16(C4○D8), C22.54(C2×D28), C2.5(D56⋊7C2), C28.44D4⋊44C2, C28.66(C22⋊C4), (C2×C56).356C22, C28.113(C22×C4), (C2×C28).767C23, C22.6(D14⋊C4), Dic14.24(C2×C4), (C22×C14).139D4, (C22×C4).429D14, C7⋊4(C23.24D4), (C2×D28).199C22, C23.21D14⋊2C2, C4⋊Dic7.282C22, (C22×C28).542C22, (C2×Dic14).219C22, C4.71(C2×C4×D7), C2.25(C2×D14⋊C4), (C2×C4○D28).5C2, (C2×C4).117(C4×D7), C4.104(C2×C7⋊D4), (C2×C28).235(C2×C4), (C2×C14).157(C2×D4), C14.53(C2×C22⋊C4), (C2×C4).255(C7⋊D4), (C2×C4).715(C22×D7), (C2×C14).64(C22⋊C4), SmallGroup(448,647)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.23D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd27 >
Subgroups: 804 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C56, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C23.24D4, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C28.44D4, C2.D56, C23.21D14, C22×C56, C2×C4○D28, C23.23D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C4○D8, C4×D7, D28, C7⋊D4, C22×D7, C23.24D4, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D56⋊7C2, C2×D14⋊C4, C23.23D28
►On 224 points
Generators in S224
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 169)(120 170)(121 171)(122 172)(123 173)(124 174)(125 175)(126 176)(127 177)(128 178)(129 179)(130 180)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)(137 187)(138 188)(139 189)(140 190)(141 191)(142 192)(143 193)(144 194)(145 195)(146 196)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 72)(42 73)(43 74)(44 75)(45 76)(46 77)(47 78)(48 79)(49 80)(50 81)(51 82)(52 83)(53 84)(54 85)(55 86)(56 87)(113 191)(114 192)(115 193)(116 194)(117 195)(118 196)(119 197)(120 198)(121 199)(122 200)(123 201)(124 202)(125 203)(126 204)(127 205)(128 206)(129 207)(130 208)(131 209)(132 210)(133 211)(134 212)(135 213)(136 214)(137 215)(138 216)(139 217)(140 218)(141 219)(142 220)(143 221)(144 222)(145 223)(146 224)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)(155 177)(156 178)(157 179)(158 180)(159 181)(160 182)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)
(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)
(1 187 60 137)(2 136 61 186)(3 185 62 135)(4 134 63 184)(5 183 64 133)(6 132 65 182)(7 181 66 131)(8 130 67 180)(9 179 68 129)(10 128 69 178)(11 177 70 127)(12 126 71 176)(13 175 72 125)(14 124 73 174)(15 173 74 123)(16 122 75 172)(17 171 76 121)(18 120 77 170)(19 169 78 119)(20 118 79 224)(21 223 80 117)(22 116 81 222)(23 221 82 115)(24 114 83 220)(25 219 84 113)(26 168 85 218)(27 217 86 167)(28 166 87 216)(29 215 88 165)(30 164 89 214)(31 213 90 163)(32 162 91 212)(33 211 92 161)(34 160 93 210)(35 209 94 159)(36 158 95 208)(37 207 96 157)(38 156 97 206)(39 205 98 155)(40 154 99 204)(41 203 100 153)(42 152 101 202)(43 201 102 151)(44 150 103 200)(45 199 104 149)(46 148 105 198)(47 197 106 147)(48 146 107 196)(49 195 108 145)(50 144 109 194)(51 193 110 143)(52 142 111 192)(53 191 112 141)(54 140 57 190)(55 189 58 139)(56 138 59 188)
G:=sub<Sym(224)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,215)(138,216)(139,217)(140,218)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(155,177)(156,178)(157,179)(158,180)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190), (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), (1,187,60,137)(2,136,61,186)(3,185,62,135)(4,134,63,184)(5,183,64,133)(6,132,65,182)(7,181,66,131)(8,130,67,180)(9,179,68,129)(10,128,69,178)(11,177,70,127)(12,126,71,176)(13,175,72,125)(14,124,73,174)(15,173,74,123)(16,122,75,172)(17,171,76,121)(18,120,77,170)(19,169,78,119)(20,118,79,224)(21,223,80,117)(22,116,81,222)(23,221,82,115)(24,114,83,220)(25,219,84,113)(26,168,85,218)(27,217,86,167)(28,166,87,216)(29,215,88,165)(30,164,89,214)(31,213,90,163)(32,162,91,212)(33,211,92,161)(34,160,93,210)(35,209,94,159)(36,158,95,208)(37,207,96,157)(38,156,97,206)(39,205,98,155)(40,154,99,204)(41,203,100,153)(42,152,101,202)(43,201,102,151)(44,150,103,200)(45,199,104,149)(46,148,105,198)(47,197,106,147)(48,146,107,196)(49,195,108,145)(50,144,109,194)(51,193,110,143)(52,142,111,192)(53,191,112,141)(54,140,57,190)(55,189,58,139)(56,138,59,188)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,169)(120,170)(121,171)(122,172)(123,173)(124,174)(125,175)(126,176)(127,177)(128,178)(129,179)(130,180)(131,181)(132,182)(133,183)(134,184)(135,185)(136,186)(137,187)(138,188)(139,189)(140,190)(141,191)(142,192)(143,193)(144,194)(145,195)(146,196)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,72)(42,73)(43,74)(44,75)(45,76)(46,77)(47,78)(48,79)(49,80)(50,81)(51,82)(52,83)(53,84)(54,85)(55,86)(56,87)(113,191)(114,192)(115,193)(116,194)(117,195)(118,196)(119,197)(120,198)(121,199)(122,200)(123,201)(124,202)(125,203)(126,204)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,215)(138,216)(139,217)(140,218)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(155,177)(156,178)(157,179)(158,180)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190), (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), (1,187,60,137)(2,136,61,186)(3,185,62,135)(4,134,63,184)(5,183,64,133)(6,132,65,182)(7,181,66,131)(8,130,67,180)(9,179,68,129)(10,128,69,178)(11,177,70,127)(12,126,71,176)(13,175,72,125)(14,124,73,174)(15,173,74,123)(16,122,75,172)(17,171,76,121)(18,120,77,170)(19,169,78,119)(20,118,79,224)(21,223,80,117)(22,116,81,222)(23,221,82,115)(24,114,83,220)(25,219,84,113)(26,168,85,218)(27,217,86,167)(28,166,87,216)(29,215,88,165)(30,164,89,214)(31,213,90,163)(32,162,91,212)(33,211,92,161)(34,160,93,210)(35,209,94,159)(36,158,95,208)(37,207,96,157)(38,156,97,206)(39,205,98,155)(40,154,99,204)(41,203,100,153)(42,152,101,202)(43,201,102,151)(44,150,103,200)(45,199,104,149)(46,148,105,198)(47,197,106,147)(48,146,107,196)(49,195,108,145)(50,144,109,194)(51,193,110,143)(52,142,111,192)(53,191,112,141)(54,140,57,190)(55,189,58,139)(56,138,59,188) );
G=PermutationGroup([[(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,169),(120,170),(121,171),(122,172),(123,173),(124,174),(125,175),(126,176),(127,177),(128,178),(129,179),(130,180),(131,181),(132,182),(133,183),(134,184),(135,185),(136,186),(137,187),(138,188),(139,189),(140,190),(141,191),(142,192),(143,193),(144,194),(145,195),(146,196),(147,197),(148,198),(149,199),(150,200),(151,201),(152,202),(153,203),(154,204),(155,205),(156,206),(157,207),(158,208),(159,209),(160,210),(161,211),(162,212),(163,213),(164,214),(165,215),(166,216),(167,217),(168,218)], [(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,72),(42,73),(43,74),(44,75),(45,76),(46,77),(47,78),(48,79),(49,80),(50,81),(51,82),(52,83),(53,84),(54,85),(55,86),(56,87),(113,191),(114,192),(115,193),(116,194),(117,195),(118,196),(119,197),(120,198),(121,199),(122,200),(123,201),(124,202),(125,203),(126,204),(127,205),(128,206),(129,207),(130,208),(131,209),(132,210),(133,211),(134,212),(135,213),(136,214),(137,215),(138,216),(139,217),(140,218),(141,219),(142,220),(143,221),(144,222),(145,223),(146,224),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176),(155,177),(156,178),(157,179),(158,180),(159,181),(160,182),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190)], [(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)], [(1,187,60,137),(2,136,61,186),(3,185,62,135),(4,134,63,184),(5,183,64,133),(6,132,65,182),(7,181,66,131),(8,130,67,180),(9,179,68,129),(10,128,69,178),(11,177,70,127),(12,126,71,176),(13,175,72,125),(14,124,73,174),(15,173,74,123),(16,122,75,172),(17,171,76,121),(18,120,77,170),(19,169,78,119),(20,118,79,224),(21,223,80,117),(22,116,81,222),(23,221,82,115),(24,114,83,220),(25,219,84,113),(26,168,85,218),(27,217,86,167),(28,166,87,216),(29,215,88,165),(30,164,89,214),(31,213,90,163),(32,162,91,212),(33,211,92,161),(34,160,93,210),(35,209,94,159),(36,158,95,208),(37,207,96,157),(38,156,97,206),(39,205,98,155),(40,154,99,204),(41,203,100,153),(42,152,101,202),(43,201,102,151),(44,150,103,200),(45,199,104,149),(46,148,105,198),(47,197,106,147),(48,146,107,196),(49,195,108,145),(50,144,109,194),(51,193,110,143),(52,142,111,192),(53,191,112,141),(54,140,57,190),(55,189,58,139),(56,138,59,188)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | C4○D8 | C4×D7 | D28 | C7⋊D4 | D28 | D56⋊7C2 |
| kernel | C23.23D28 | C28.44D4 | C2.D56 | C23.21D14 | C22×C56 | C2×C4○D28 | C4○D28 | C2×C28 | C22×C14 | C22×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 6 | 3 | 8 | 12 | 6 | 12 | 6 | 48 |
Matrix representation of C23.23D28 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 47 | 112 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 98 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 37 | 9 |
| 15 | 0 | 0 |
| 0 | 99 | 3 |
| 0 | 85 | 14 |
G:=sub<GL(3,GF(113))| [1,0,0,0,1,47,0,0,112],[112,0,0,0,1,0,0,0,1],[1,0,0,0,112,0,0,0,112],[98,0,0,0,25,37,0,0,9],[15,0,0,0,99,85,0,3,14] >;
C23.23D28 in GAP, Magma, Sage, TeX
C_2^3._{23}D_{28} % in TeX
G:=Group("C2^3.23D28"); // GroupNames label
G:=SmallGroup(448,647);
// by ID
G=gap.SmallGroup(448,647);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,142,1123,136,18822]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^27>;
// generators/relations