metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C28).56D4, (C2×C4).45D28, (C22×D7).5Q8, C22.51(Q8×D7), (C2×Dic7).62D4, C22.250(D4×D7), C2.11(C28⋊7D4), C14.63(C4⋊D4), C2.26(C28⋊1D4), C7⋊3(C23.Q8), C2.9(D14⋊3Q8), (C22×C4).106D14, C22.130(C2×D28), C14.51(C22⋊Q8), C14.C42⋊21C2, C2.17(D14⋊2Q8), C2.23(D14⋊Q8), (C22×C28).69C22, (C23×D7).23C22, C23.383(C22×D7), C14.30(C42⋊2C2), C2.13(Dic7⋊D4), C22.111(C4○D28), (C22×C14).358C23, C22.54(Q8⋊2D7), C22.105(D4⋊2D7), (C22×Dic7).63C22, (C2×C4⋊C4)⋊12D7, (C14×C4⋊C4)⋊23C2, (C2×C4⋊Dic7)⋊14C2, (C2×C14).86(C2×Q8), (C2×Dic7⋊C4)⋊42C2, (C2×D14⋊C4).24C2, (C2×C14).338(C2×D4), (C2×C4).44(C7⋊D4), C2.15(C4⋊C4⋊D7), (C2×C14).87(C4○D4), C22.143(C2×C7⋊D4), SmallGroup(448,528)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C28).56D4
G = < a,b,c,d | a2=b28=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=ac-1 >
Subgroups: 996 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×9], C22 [×7], C22 [×10], C7, C2×C4 [×4], C2×C4 [×17], C23, C23 [×8], D7 [×2], C14 [×7], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, Dic7 [×4], C28 [×5], D14 [×10], C2×C14 [×7], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×Dic7 [×2], C2×Dic7 [×8], C2×C28 [×4], C2×C28 [×7], C22×D7 [×2], C22×D7 [×6], C22×C14, C23.Q8, Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4 [×6], C7×C4⋊C4 [×2], C22×Dic7 [×3], C22×C28 [×3], C23×D7, C14.C42, C2×Dic7⋊C4, C2×C4⋊Dic7, C2×D14⋊C4 [×3], C14×C4⋊C4, (C2×C28).56D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D7, C2×D4 [×3], C2×Q8, C4○D4 [×3], D14 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C42⋊2C2, D28 [×2], C7⋊D4 [×2], C22×D7, C23.Q8, C2×D28, C4○D28, D4×D7, D4⋊2D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, C28⋊1D4, D14⋊Q8, D14⋊2Q8, C4⋊C4⋊D7, C28⋊7D4, Dic7⋊D4, D14⋊3Q8, (C2×C28).56D4
►On 224 points
Generators in S224
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 216)(30 217)(31 218)(32 219)(33 220)(34 221)(35 222)(36 223)(37 224)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 211)(53 212)(54 213)(55 214)(56 215)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(141 186)(142 187)(143 188)(144 189)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 196)(152 169)(153 170)(154 171)(155 172)(156 173)(157 174)(158 175)(159 176)(160 177)(161 178)(162 179)(163 180)(164 181)(165 182)(166 183)(167 184)(168 185)
(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)
(1 146 54 129)(2 145 55 128)(3 144 56 127)(4 143 29 126)(5 142 30 125)(6 141 31 124)(7 168 32 123)(8 167 33 122)(9 166 34 121)(10 165 35 120)(11 164 36 119)(12 163 37 118)(13 162 38 117)(14 161 39 116)(15 160 40 115)(16 159 41 114)(17 158 42 113)(18 157 43 140)(19 156 44 139)(20 155 45 138)(21 154 46 137)(22 153 47 136)(23 152 48 135)(24 151 49 134)(25 150 50 133)(26 149 51 132)(27 148 52 131)(28 147 53 130)(57 184 220 101)(58 183 221 100)(59 182 222 99)(60 181 223 98)(61 180 224 97)(62 179 197 96)(63 178 198 95)(64 177 199 94)(65 176 200 93)(66 175 201 92)(67 174 202 91)(68 173 203 90)(69 172 204 89)(70 171 205 88)(71 170 206 87)(72 169 207 86)(73 196 208 85)(74 195 209 112)(75 194 210 111)(76 193 211 110)(77 192 212 109)(78 191 213 108)(79 190 214 107)(80 189 215 106)(81 188 216 105)(82 187 217 104)(83 186 218 103)(84 185 219 102)
(1 78)(2 28)(3 76)(4 26)(5 74)(6 24)(7 72)(8 22)(9 70)(10 20)(11 68)(12 18)(13 66)(14 16)(15 64)(17 62)(19 60)(21 58)(23 84)(25 82)(27 80)(29 51)(30 209)(31 49)(32 207)(33 47)(34 205)(35 45)(36 203)(37 43)(38 201)(39 41)(40 199)(42 197)(44 223)(46 221)(48 219)(50 217)(52 215)(53 55)(54 213)(56 211)(57 71)(59 69)(61 67)(63 65)(73 83)(75 81)(77 79)(85 141)(86 185)(87 167)(88 183)(89 165)(90 181)(91 163)(92 179)(93 161)(94 177)(95 159)(96 175)(97 157)(98 173)(99 155)(100 171)(101 153)(102 169)(103 151)(104 195)(105 149)(106 193)(107 147)(108 191)(109 145)(110 189)(111 143)(112 187)(113 162)(114 178)(115 160)(116 176)(117 158)(118 174)(119 156)(120 172)(121 154)(122 170)(123 152)(124 196)(125 150)(126 194)(127 148)(128 192)(129 146)(130 190)(131 144)(132 188)(133 142)(134 186)(135 168)(136 184)(137 166)(138 182)(139 164)(140 180)(198 200)(202 224)(204 222)(206 220)(208 218)(210 216)(212 214)
G:=sub<Sym(224)| (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(141,186)(142,187)(143,188)(144,189)(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)(152,169)(153,170)(154,171)(155,172)(156,173)(157,174)(158,175)(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)(166,183)(167,184)(168,185), (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), (1,146,54,129)(2,145,55,128)(3,144,56,127)(4,143,29,126)(5,142,30,125)(6,141,31,124)(7,168,32,123)(8,167,33,122)(9,166,34,121)(10,165,35,120)(11,164,36,119)(12,163,37,118)(13,162,38,117)(14,161,39,116)(15,160,40,115)(16,159,41,114)(17,158,42,113)(18,157,43,140)(19,156,44,139)(20,155,45,138)(21,154,46,137)(22,153,47,136)(23,152,48,135)(24,151,49,134)(25,150,50,133)(26,149,51,132)(27,148,52,131)(28,147,53,130)(57,184,220,101)(58,183,221,100)(59,182,222,99)(60,181,223,98)(61,180,224,97)(62,179,197,96)(63,178,198,95)(64,177,199,94)(65,176,200,93)(66,175,201,92)(67,174,202,91)(68,173,203,90)(69,172,204,89)(70,171,205,88)(71,170,206,87)(72,169,207,86)(73,196,208,85)(74,195,209,112)(75,194,210,111)(76,193,211,110)(77,192,212,109)(78,191,213,108)(79,190,214,107)(80,189,215,106)(81,188,216,105)(82,187,217,104)(83,186,218,103)(84,185,219,102), (1,78)(2,28)(3,76)(4,26)(5,74)(6,24)(7,72)(8,22)(9,70)(10,20)(11,68)(12,18)(13,66)(14,16)(15,64)(17,62)(19,60)(21,58)(23,84)(25,82)(27,80)(29,51)(30,209)(31,49)(32,207)(33,47)(34,205)(35,45)(36,203)(37,43)(38,201)(39,41)(40,199)(42,197)(44,223)(46,221)(48,219)(50,217)(52,215)(53,55)(54,213)(56,211)(57,71)(59,69)(61,67)(63,65)(73,83)(75,81)(77,79)(85,141)(86,185)(87,167)(88,183)(89,165)(90,181)(91,163)(92,179)(93,161)(94,177)(95,159)(96,175)(97,157)(98,173)(99,155)(100,171)(101,153)(102,169)(103,151)(104,195)(105,149)(106,193)(107,147)(108,191)(109,145)(110,189)(111,143)(112,187)(113,162)(114,178)(115,160)(116,176)(117,158)(118,174)(119,156)(120,172)(121,154)(122,170)(123,152)(124,196)(125,150)(126,194)(127,148)(128,192)(129,146)(130,190)(131,144)(132,188)(133,142)(134,186)(135,168)(136,184)(137,166)(138,182)(139,164)(140,180)(198,200)(202,224)(204,222)(206,220)(208,218)(210,216)(212,214)>;
G:=Group( (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,216)(30,217)(31,218)(32,219)(33,220)(34,221)(35,222)(36,223)(37,224)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(85,134)(86,135)(87,136)(88,137)(89,138)(90,139)(91,140)(92,113)(93,114)(94,115)(95,116)(96,117)(97,118)(98,119)(99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(141,186)(142,187)(143,188)(144,189)(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)(152,169)(153,170)(154,171)(155,172)(156,173)(157,174)(158,175)(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)(166,183)(167,184)(168,185), (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), (1,146,54,129)(2,145,55,128)(3,144,56,127)(4,143,29,126)(5,142,30,125)(6,141,31,124)(7,168,32,123)(8,167,33,122)(9,166,34,121)(10,165,35,120)(11,164,36,119)(12,163,37,118)(13,162,38,117)(14,161,39,116)(15,160,40,115)(16,159,41,114)(17,158,42,113)(18,157,43,140)(19,156,44,139)(20,155,45,138)(21,154,46,137)(22,153,47,136)(23,152,48,135)(24,151,49,134)(25,150,50,133)(26,149,51,132)(27,148,52,131)(28,147,53,130)(57,184,220,101)(58,183,221,100)(59,182,222,99)(60,181,223,98)(61,180,224,97)(62,179,197,96)(63,178,198,95)(64,177,199,94)(65,176,200,93)(66,175,201,92)(67,174,202,91)(68,173,203,90)(69,172,204,89)(70,171,205,88)(71,170,206,87)(72,169,207,86)(73,196,208,85)(74,195,209,112)(75,194,210,111)(76,193,211,110)(77,192,212,109)(78,191,213,108)(79,190,214,107)(80,189,215,106)(81,188,216,105)(82,187,217,104)(83,186,218,103)(84,185,219,102), (1,78)(2,28)(3,76)(4,26)(5,74)(6,24)(7,72)(8,22)(9,70)(10,20)(11,68)(12,18)(13,66)(14,16)(15,64)(17,62)(19,60)(21,58)(23,84)(25,82)(27,80)(29,51)(30,209)(31,49)(32,207)(33,47)(34,205)(35,45)(36,203)(37,43)(38,201)(39,41)(40,199)(42,197)(44,223)(46,221)(48,219)(50,217)(52,215)(53,55)(54,213)(56,211)(57,71)(59,69)(61,67)(63,65)(73,83)(75,81)(77,79)(85,141)(86,185)(87,167)(88,183)(89,165)(90,181)(91,163)(92,179)(93,161)(94,177)(95,159)(96,175)(97,157)(98,173)(99,155)(100,171)(101,153)(102,169)(103,151)(104,195)(105,149)(106,193)(107,147)(108,191)(109,145)(110,189)(111,143)(112,187)(113,162)(114,178)(115,160)(116,176)(117,158)(118,174)(119,156)(120,172)(121,154)(122,170)(123,152)(124,196)(125,150)(126,194)(127,148)(128,192)(129,146)(130,190)(131,144)(132,188)(133,142)(134,186)(135,168)(136,184)(137,166)(138,182)(139,164)(140,180)(198,200)(202,224)(204,222)(206,220)(208,218)(210,216)(212,214) );
G=PermutationGroup([(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,216),(30,217),(31,218),(32,219),(33,220),(34,221),(35,222),(36,223),(37,224),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,208),(50,209),(51,210),(52,211),(53,212),(54,213),(55,214),(56,215),(85,134),(86,135),(87,136),(88,137),(89,138),(90,139),(91,140),(92,113),(93,114),(94,115),(95,116),(96,117),(97,118),(98,119),(99,120),(100,121),(101,122),(102,123),(103,124),(104,125),(105,126),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(141,186),(142,187),(143,188),(144,189),(145,190),(146,191),(147,192),(148,193),(149,194),(150,195),(151,196),(152,169),(153,170),(154,171),(155,172),(156,173),(157,174),(158,175),(159,176),(160,177),(161,178),(162,179),(163,180),(164,181),(165,182),(166,183),(167,184),(168,185)], [(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)], [(1,146,54,129),(2,145,55,128),(3,144,56,127),(4,143,29,126),(5,142,30,125),(6,141,31,124),(7,168,32,123),(8,167,33,122),(9,166,34,121),(10,165,35,120),(11,164,36,119),(12,163,37,118),(13,162,38,117),(14,161,39,116),(15,160,40,115),(16,159,41,114),(17,158,42,113),(18,157,43,140),(19,156,44,139),(20,155,45,138),(21,154,46,137),(22,153,47,136),(23,152,48,135),(24,151,49,134),(25,150,50,133),(26,149,51,132),(27,148,52,131),(28,147,53,130),(57,184,220,101),(58,183,221,100),(59,182,222,99),(60,181,223,98),(61,180,224,97),(62,179,197,96),(63,178,198,95),(64,177,199,94),(65,176,200,93),(66,175,201,92),(67,174,202,91),(68,173,203,90),(69,172,204,89),(70,171,205,88),(71,170,206,87),(72,169,207,86),(73,196,208,85),(74,195,209,112),(75,194,210,111),(76,193,211,110),(77,192,212,109),(78,191,213,108),(79,190,214,107),(80,189,215,106),(81,188,216,105),(82,187,217,104),(83,186,218,103),(84,185,219,102)], [(1,78),(2,28),(3,76),(4,26),(5,74),(6,24),(7,72),(8,22),(9,70),(10,20),(11,68),(12,18),(13,66),(14,16),(15,64),(17,62),(19,60),(21,58),(23,84),(25,82),(27,80),(29,51),(30,209),(31,49),(32,207),(33,47),(34,205),(35,45),(36,203),(37,43),(38,201),(39,41),(40,199),(42,197),(44,223),(46,221),(48,219),(50,217),(52,215),(53,55),(54,213),(56,211),(57,71),(59,69),(61,67),(63,65),(73,83),(75,81),(77,79),(85,141),(86,185),(87,167),(88,183),(89,165),(90,181),(91,163),(92,179),(93,161),(94,177),(95,159),(96,175),(97,157),(98,173),(99,155),(100,171),(101,153),(102,169),(103,151),(104,195),(105,149),(106,193),(107,147),(108,191),(109,145),(110,189),(111,143),(112,187),(113,162),(114,178),(115,160),(116,176),(117,158),(118,174),(119,156),(120,172),(121,154),(122,170),(123,152),(124,196),(125,150),(126,194),(127,148),(128,192),(129,146),(130,190),(131,144),(132,188),(133,142),(134,186),(135,168),(136,184),(137,166),(138,182),(139,164),(140,180),(198,200),(202,224),(204,222),(206,220),(208,218),(210,216),(212,214)])
82 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 28 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | Q8 | D7 | C4○D4 | D14 | D28 | C7⋊D4 | C4○D28 | D4×D7 | D4⋊2D7 | Q8×D7 | Q8⋊2D7 |
| kernel | (C2×C28).56D4 | C14.C42 | C2×Dic7⋊C4 | C2×C4⋊Dic7 | C2×D14⋊C4 | C14×C4⋊C4 | C2×Dic7 | C2×C28 | C22×D7 | C2×C4⋊C4 | C2×C14 | C22×C4 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 4 | 2 | 3 | 6 | 9 | 12 | 12 | 12 | 3 | 3 | 3 | 3 |
Matrix representation of (C2×C28).56D4 ►in GL6(𝔽29)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 14 | 8 | 0 | 0 | 0 | 0 |
| 8 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 24 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 0 | 24 | 17 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 13 |
| 0 | 0 | 0 | 0 | 27 | 5 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,8,0,0,0,0,8,15,0,0,0,0,0,0,27,5,0,0,0,0,24,12,0,0,0,0,0,0,2,24,0,0,0,0,5,17],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,24,27,0,0,0,0,13,5],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,1,3,0,0,0,0,0,28] >;
(C2×C28).56D4 in GAP, Magma, Sage, TeX
(C_2\times C_{28})._{56}D_4 % in TeX
G:=Group("(C2xC28).56D4"); // GroupNames label
G:=SmallGroup(448,528);
// by ID
G=gap.SmallGroup(448,528);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,387,184,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a*b^-1,d*c*d=a*c^-1>;
// generators/relations