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## G = C2×C22.D28order 448 = 26·7

### Direct product of C2 and C22.D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C22.D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×C7⋊D4 — C2×C22.D28
 Lower central C7 — C2×C14 — C2×C22.D28
 Upper central C1 — C23 — C2×C22⋊C4

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, D42D7, C23×D7, C22.D28, C22×D28, C2×D42D7, C2×C22.D28

Smallest permutation representation of 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

׿
×
𝔽