direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.D28, C23.51D28, C24.55D14, C22⋊C4⋊42D14, C2.9(C22×D28), C14.7(C22×D4), D14⋊C4⋊48C22, (C2×C14).36C24, C4⋊Dic7⋊52C22, (C23×Dic7)⋊4C2, C22.66(C2×D28), (C2×C28).129C23, (C22×C4).171D14, (C22×C14).117D4, (C2×Dic7).9C23, (C22×D7).8C23, C22.75(C23×D7), C14⋊2(C22.D4), (C22×C28).72C22, (C23×C14).62C22, (C23×D7).33C22, C23.147(C22×D7), C22.69(D4⋊2D7), (C22×C14).126C23, (C22×Dic7)⋊42C22, (C2×D14⋊C4)⋊19C2, (C2×C4⋊Dic7)⋊20C2, (C2×C22⋊C4)⋊15D7, C14.68(C2×C4○D4), (C2×C14).48(C2×D4), (C14×C22⋊C4)⋊14C2, C7⋊2(C2×C22.D4), C2.11(C2×D4⋊2D7), (C7×C22⋊C4)⋊47C22, (C2×C4).135(C22×D7), (C2×C7⋊D4).91C22, (C22×C7⋊D4).12C2, (C2×C14).168(C4○D4), SmallGroup(448,945)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 1556 in 342 conjugacy classes, 127 normal (17 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, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22.D4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C22.D28, C2×C4⋊Dic7, C2×D14⋊C4, C14×C22⋊C4, C23×Dic7, C22×C7⋊D4, C2×C22.D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, D28, C22×D7, C2×C22.D4, C2×D28, D4⋊2D7, C23×D7, C22.D28, C22×D28, C2×D4⋊2D7, C2×C22.D28
►On 224 points
Generators in S224
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(113 178)(114 179)(115 180)(116 181)(117 182)(118 183)(119 184)(120 185)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 193)(129 194)(130 195)(131 196)(132 169)(133 170)(134 171)(135 172)(136 173)(137 174)(138 175)(139 176)(140 177)(141 210)(142 211)(143 212)(144 213)(145 214)(146 215)(147 216)(148 217)(149 218)(150 219)(151 220)(152 221)(153 222)(154 223)(155 224)(156 197)(157 198)(158 199)(159 200)(160 201)(161 202)(162 203)(163 204)(164 205)(165 206)(166 207)(167 208)(168 209)
(1 171)(2 37)(3 173)(4 39)(5 175)(6 41)(7 177)(8 43)(9 179)(10 45)(11 181)(12 47)(13 183)(14 49)(15 185)(16 51)(17 187)(18 53)(19 189)(20 55)(21 191)(22 29)(23 193)(24 31)(25 195)(26 33)(27 169)(28 35)(30 114)(32 116)(34 118)(36 120)(38 122)(40 124)(42 126)(44 128)(46 130)(48 132)(50 134)(52 136)(54 138)(56 140)(57 218)(58 85)(59 220)(60 87)(61 222)(62 89)(63 224)(64 91)(65 198)(66 93)(67 200)(68 95)(69 202)(70 97)(71 204)(72 99)(73 206)(74 101)(75 208)(76 103)(77 210)(78 105)(79 212)(80 107)(81 214)(82 109)(83 216)(84 111)(86 165)(88 167)(90 141)(92 143)(94 145)(96 147)(98 149)(100 151)(102 153)(104 155)(106 157)(108 159)(110 161)(112 163)(113 192)(115 194)(117 196)(119 170)(121 172)(123 174)(125 176)(127 178)(129 180)(131 182)(133 184)(135 186)(137 188)(139 190)(142 197)(144 199)(146 201)(148 203)(150 205)(152 207)(154 209)(156 211)(158 213)(160 215)(162 217)(164 219)(166 221)(168 223)
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 192)(30 193)(31 194)(32 195)(33 196)(34 169)(35 170)(36 171)(37 172)(38 173)(39 174)(40 175)(41 176)(42 177)(43 178)(44 179)(45 180)(46 181)(47 182)(48 183)(49 184)(50 185)(51 186)(52 187)(53 188)(54 189)(55 190)(56 191)(57 163)(58 164)(59 165)(60 166)(61 167)(62 168)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 151)(74 152)(75 153)(76 154)(77 155)(78 156)(79 157)(80 158)(81 159)(82 160)(83 161)(84 162)(85 219)(86 220)(87 221)(88 222)(89 223)(90 224)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 211)(106 212)(107 213)(108 214)(109 215)(110 216)(111 217)(112 218)
(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)
(1 204 120 98)(2 97 121 203)(3 202 122 96)(4 95 123 201)(5 200 124 94)(6 93 125 199)(7 198 126 92)(8 91 127 197)(9 224 128 90)(10 89 129 223)(11 222 130 88)(12 87 131 221)(13 220 132 86)(14 85 133 219)(15 218 134 112)(16 111 135 217)(17 216 136 110)(18 109 137 215)(19 214 138 108)(20 107 139 213)(21 212 140 106)(22 105 113 211)(23 210 114 104)(24 103 115 209)(25 208 116 102)(26 101 117 207)(27 206 118 100)(28 99 119 205)(29 78 192 156)(30 155 193 77)(31 76 194 154)(32 153 195 75)(33 74 196 152)(34 151 169 73)(35 72 170 150)(36 149 171 71)(37 70 172 148)(38 147 173 69)(39 68 174 146)(40 145 175 67)(41 66 176 144)(42 143 177 65)(43 64 178 142)(44 141 179 63)(45 62 180 168)(46 167 181 61)(47 60 182 166)(48 165 183 59)(49 58 184 164)(50 163 185 57)(51 84 186 162)(52 161 187 83)(53 82 188 160)(54 159 189 81)(55 80 190 158)(56 157 191 79)
G:=sub<Sym(224)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,169)(133,170)(134,171)(135,172)(136,173)(137,174)(138,175)(139,176)(140,177)(141,210)(142,211)(143,212)(144,213)(145,214)(146,215)(147,216)(148,217)(149,218)(150,219)(151,220)(152,221)(153,222)(154,223)(155,224)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209), (1,171)(2,37)(3,173)(4,39)(5,175)(6,41)(7,177)(8,43)(9,179)(10,45)(11,181)(12,47)(13,183)(14,49)(15,185)(16,51)(17,187)(18,53)(19,189)(20,55)(21,191)(22,29)(23,193)(24,31)(25,195)(26,33)(27,169)(28,35)(30,114)(32,116)(34,118)(36,120)(38,122)(40,124)(42,126)(44,128)(46,130)(48,132)(50,134)(52,136)(54,138)(56,140)(57,218)(58,85)(59,220)(60,87)(61,222)(62,89)(63,224)(64,91)(65,198)(66,93)(67,200)(68,95)(69,202)(70,97)(71,204)(72,99)(73,206)(74,101)(75,208)(76,103)(77,210)(78,105)(79,212)(80,107)(81,214)(82,109)(83,216)(84,111)(86,165)(88,167)(90,141)(92,143)(94,145)(96,147)(98,149)(100,151)(102,153)(104,155)(106,157)(108,159)(110,161)(112,163)(113,192)(115,194)(117,196)(119,170)(121,172)(123,174)(125,176)(127,178)(129,180)(131,182)(133,184)(135,186)(137,188)(139,190)(142,197)(144,199)(146,201)(148,203)(150,205)(152,207)(154,209)(156,211)(158,213)(160,215)(162,217)(164,219)(166,221)(168,223), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,192)(30,193)(31,194)(32,195)(33,196)(34,169)(35,170)(36,171)(37,172)(38,173)(39,174)(40,175)(41,176)(42,177)(43,178)(44,179)(45,180)(46,181)(47,182)(48,183)(49,184)(50,185)(51,186)(52,187)(53,188)(54,189)(55,190)(56,191)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,219)(86,220)(87,221)(88,222)(89,223)(90,224)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218), (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), (1,204,120,98)(2,97,121,203)(3,202,122,96)(4,95,123,201)(5,200,124,94)(6,93,125,199)(7,198,126,92)(8,91,127,197)(9,224,128,90)(10,89,129,223)(11,222,130,88)(12,87,131,221)(13,220,132,86)(14,85,133,219)(15,218,134,112)(16,111,135,217)(17,216,136,110)(18,109,137,215)(19,214,138,108)(20,107,139,213)(21,212,140,106)(22,105,113,211)(23,210,114,104)(24,103,115,209)(25,208,116,102)(26,101,117,207)(27,206,118,100)(28,99,119,205)(29,78,192,156)(30,155,193,77)(31,76,194,154)(32,153,195,75)(33,74,196,152)(34,151,169,73)(35,72,170,150)(36,149,171,71)(37,70,172,148)(38,147,173,69)(39,68,174,146)(40,145,175,67)(41,66,176,144)(42,143,177,65)(43,64,178,142)(44,141,179,63)(45,62,180,168)(46,167,181,61)(47,60,182,166)(48,165,183,59)(49,58,184,164)(50,163,185,57)(51,84,186,162)(52,161,187,83)(53,82,188,160)(54,159,189,81)(55,80,190,158)(56,157,191,79)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,169)(133,170)(134,171)(135,172)(136,173)(137,174)(138,175)(139,176)(140,177)(141,210)(142,211)(143,212)(144,213)(145,214)(146,215)(147,216)(148,217)(149,218)(150,219)(151,220)(152,221)(153,222)(154,223)(155,224)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209), (1,171)(2,37)(3,173)(4,39)(5,175)(6,41)(7,177)(8,43)(9,179)(10,45)(11,181)(12,47)(13,183)(14,49)(15,185)(16,51)(17,187)(18,53)(19,189)(20,55)(21,191)(22,29)(23,193)(24,31)(25,195)(26,33)(27,169)(28,35)(30,114)(32,116)(34,118)(36,120)(38,122)(40,124)(42,126)(44,128)(46,130)(48,132)(50,134)(52,136)(54,138)(56,140)(57,218)(58,85)(59,220)(60,87)(61,222)(62,89)(63,224)(64,91)(65,198)(66,93)(67,200)(68,95)(69,202)(70,97)(71,204)(72,99)(73,206)(74,101)(75,208)(76,103)(77,210)(78,105)(79,212)(80,107)(81,214)(82,109)(83,216)(84,111)(86,165)(88,167)(90,141)(92,143)(94,145)(96,147)(98,149)(100,151)(102,153)(104,155)(106,157)(108,159)(110,161)(112,163)(113,192)(115,194)(117,196)(119,170)(121,172)(123,174)(125,176)(127,178)(129,180)(131,182)(133,184)(135,186)(137,188)(139,190)(142,197)(144,199)(146,201)(148,203)(150,205)(152,207)(154,209)(156,211)(158,213)(160,215)(162,217)(164,219)(166,221)(168,223), (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,192)(30,193)(31,194)(32,195)(33,196)(34,169)(35,170)(36,171)(37,172)(38,173)(39,174)(40,175)(41,176)(42,177)(43,178)(44,179)(45,180)(46,181)(47,182)(48,183)(49,184)(50,185)(51,186)(52,187)(53,188)(54,189)(55,190)(56,191)(57,163)(58,164)(59,165)(60,166)(61,167)(62,168)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,151)(74,152)(75,153)(76,154)(77,155)(78,156)(79,157)(80,158)(81,159)(82,160)(83,161)(84,162)(85,219)(86,220)(87,221)(88,222)(89,223)(90,224)(91,197)(92,198)(93,199)(94,200)(95,201)(96,202)(97,203)(98,204)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,211)(106,212)(107,213)(108,214)(109,215)(110,216)(111,217)(112,218), (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), (1,204,120,98)(2,97,121,203)(3,202,122,96)(4,95,123,201)(5,200,124,94)(6,93,125,199)(7,198,126,92)(8,91,127,197)(9,224,128,90)(10,89,129,223)(11,222,130,88)(12,87,131,221)(13,220,132,86)(14,85,133,219)(15,218,134,112)(16,111,135,217)(17,216,136,110)(18,109,137,215)(19,214,138,108)(20,107,139,213)(21,212,140,106)(22,105,113,211)(23,210,114,104)(24,103,115,209)(25,208,116,102)(26,101,117,207)(27,206,118,100)(28,99,119,205)(29,78,192,156)(30,155,193,77)(31,76,194,154)(32,153,195,75)(33,74,196,152)(34,151,169,73)(35,72,170,150)(36,149,171,71)(37,70,172,148)(38,147,173,69)(39,68,174,146)(40,145,175,67)(41,66,176,144)(42,143,177,65)(43,64,178,142)(44,141,179,63)(45,62,180,168)(46,167,181,61)(47,60,182,166)(48,165,183,59)(49,58,184,164)(50,163,185,57)(51,84,186,162)(52,161,187,83)(53,82,188,160)(54,159,189,81)(55,80,190,158)(56,157,191,79) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(113,178),(114,179),(115,180),(116,181),(117,182),(118,183),(119,184),(120,185),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,193),(129,194),(130,195),(131,196),(132,169),(133,170),(134,171),(135,172),(136,173),(137,174),(138,175),(139,176),(140,177),(141,210),(142,211),(143,212),(144,213),(145,214),(146,215),(147,216),(148,217),(149,218),(150,219),(151,220),(152,221),(153,222),(154,223),(155,224),(156,197),(157,198),(158,199),(159,200),(160,201),(161,202),(162,203),(163,204),(164,205),(165,206),(166,207),(167,208),(168,209)], [(1,171),(2,37),(3,173),(4,39),(5,175),(6,41),(7,177),(8,43),(9,179),(10,45),(11,181),(12,47),(13,183),(14,49),(15,185),(16,51),(17,187),(18,53),(19,189),(20,55),(21,191),(22,29),(23,193),(24,31),(25,195),(26,33),(27,169),(28,35),(30,114),(32,116),(34,118),(36,120),(38,122),(40,124),(42,126),(44,128),(46,130),(48,132),(50,134),(52,136),(54,138),(56,140),(57,218),(58,85),(59,220),(60,87),(61,222),(62,89),(63,224),(64,91),(65,198),(66,93),(67,200),(68,95),(69,202),(70,97),(71,204),(72,99),(73,206),(74,101),(75,208),(76,103),(77,210),(78,105),(79,212),(80,107),(81,214),(82,109),(83,216),(84,111),(86,165),(88,167),(90,141),(92,143),(94,145),(96,147),(98,149),(100,151),(102,153),(104,155),(106,157),(108,159),(110,161),(112,163),(113,192),(115,194),(117,196),(119,170),(121,172),(123,174),(125,176),(127,178),(129,180),(131,182),(133,184),(135,186),(137,188),(139,190),(142,197),(144,199),(146,201),(148,203),(150,205),(152,207),(154,209),(156,211),(158,213),(160,215),(162,217),(164,219),(166,221),(168,223)], [(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,192),(30,193),(31,194),(32,195),(33,196),(34,169),(35,170),(36,171),(37,172),(38,173),(39,174),(40,175),(41,176),(42,177),(43,178),(44,179),(45,180),(46,181),(47,182),(48,183),(49,184),(50,185),(51,186),(52,187),(53,188),(54,189),(55,190),(56,191),(57,163),(58,164),(59,165),(60,166),(61,167),(62,168),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,151),(74,152),(75,153),(76,154),(77,155),(78,156),(79,157),(80,158),(81,159),(82,160),(83,161),(84,162),(85,219),(86,220),(87,221),(88,222),(89,223),(90,224),(91,197),(92,198),(93,199),(94,200),(95,201),(96,202),(97,203),(98,204),(99,205),(100,206),(101,207),(102,208),(103,209),(104,210),(105,211),(106,212),(107,213),(108,214),(109,215),(110,216),(111,217),(112,218)], [(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)], [(1,204,120,98),(2,97,121,203),(3,202,122,96),(4,95,123,201),(5,200,124,94),(6,93,125,199),(7,198,126,92),(8,91,127,197),(9,224,128,90),(10,89,129,223),(11,222,130,88),(12,87,131,221),(13,220,132,86),(14,85,133,219),(15,218,134,112),(16,111,135,217),(17,216,136,110),(18,109,137,215),(19,214,138,108),(20,107,139,213),(21,212,140,106),(22,105,113,211),(23,210,114,104),(24,103,115,209),(25,208,116,102),(26,101,117,207),(27,206,118,100),(28,99,119,205),(29,78,192,156),(30,155,193,77),(31,76,194,154),(32,153,195,75),(33,74,196,152),(34,151,169,73),(35,72,170,150),(36,149,171,71),(37,70,172,148),(38,147,173,69),(39,68,174,146),(40,145,175,67),(41,66,176,144),(42,143,177,65),(43,64,178,142),(44,141,179,63),(45,62,180,168),(46,167,181,61),(47,60,182,166),(48,165,183,59),(49,58,184,164),(50,163,185,57),(51,84,186,162),(52,161,187,83),(53,82,188,160),(54,159,189,81),(55,80,190,158),(56,157,191,79)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 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 | 2 | 2 | 2 | 2 | 28 | 28 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D28 | D4⋊2D7 |
| kernel | C2×C22.D28 | C22.D28 | C2×C4⋊Dic7 | C2×D14⋊C4 | C14×C22⋊C4 | C23×Dic7 | C22×C7⋊D4 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 8 | 12 | 6 | 3 | 24 | 12 |
Matrix representation of C2×C22.D28 ►in GL6(𝔽29)
| 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 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 | 2 | 28 |
| 1 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 20 | 4 | 0 | 0 | 0 | 0 |
| 25 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 16 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 0 | 0 | 2 | 28 |
| 22 | 28 | 0 | 0 | 0 | 0 |
| 19 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 7 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(29))| [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],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,2,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[20,25,0,0,0,0,4,21,0,0,0,0,0,0,22,16,0,0,0,0,0,4,0,0,0,0,0,0,1,2,0,0,0,0,28,28],[22,19,0,0,0,0,28,7,0,0,0,0,0,0,26,3,0,0,0,0,7,3,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
C2×C22.D28 in GAP, Magma, Sage, TeX
C_2\times C_2^2.D_{28} % in TeX
G:=Group("C2xC2^2.D28"); // GroupNames label
G:=SmallGroup(448,945);
// by ID
G=gap.SmallGroup(448,945);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,297,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations