metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D28, C42.112D14, C14.612- (1+4), (C4×D4)⋊17D7, (C7×D4)⋊11D4, (C4×D28)⋊31C2, (D4×C28)⋊19C2, C7⋊3(D4⋊6D4), C4.23(C2×D28), C28.55(C2×D4), C28⋊14(C4○D4), C28⋊7D4⋊10C2, C4⋊5(D4⋊2D7), C4⋊C4.284D14, C28⋊2Q8⋊25C2, C22.2(C2×D28), D14⋊2Q8⋊15C2, (C2×D4).249D14, (C2×C14).99C24, D14⋊C4.5C22, C2.19(C22×D28), C14.17(C22×D4), (C4×C28).155C22, (C2×C28).160C23, C22⋊C4.113D14, C22.D28⋊6C2, (C22×C4).211D14, C4⋊Dic7.39C22, (C2×D28).212C22, (D4×C14).260C22, (C22×C28).81C22, (C22×D7).34C23, C22.124(C23×D7), C23.173(C22×D7), (C22×C14).169C23, (C2×Dic7).206C23, C2.18(D4.10D14), (C2×Dic14).144C22, (C22×Dic7).97C22, (C2×C14).2(C2×D4), (C2×D4⋊2D7)⋊4C2, (C2×C4⋊Dic7)⋊25C2, C14.74(C2×C4○D4), (C2×C4×D7).65C22, C2.22(C2×D4⋊2D7), (C7×C4⋊C4).329C22, (C2×C4).732(C22×D7), (C2×C7⋊D4).15C22, (C7×C22⋊C4).106C22, SmallGroup(448,1008)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1332 in 292 conjugacy classes, 115 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×10], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×22], D4 [×4], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8 [×2], C4○D4 [×8], Dic7 [×6], C28 [×4], C28 [×3], D14 [×6], C2×C14, C2×C14 [×4], C2×C14 [×4], C2×C4⋊C4 [×2], C4×D4, C4×D4, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], D28 [×2], C2×Dic7 [×6], C2×Dic7 [×8], C7⋊D4 [×8], C2×C28 [×3], C2×C28 [×2], C2×C28 [×4], C7×D4 [×4], C22×D7 [×2], C22×C14 [×2], D4⋊6D4, C4⋊Dic7, C4⋊Dic7 [×8], D14⋊C4 [×6], C4×C28, C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, D4⋊2D7 [×8], C22×Dic7 [×4], C2×C7⋊D4 [×4], C22×C28 [×2], D4×C14, C28⋊2Q8, C4×D28, C22.D28 [×4], D14⋊2Q8 [×2], C2×C4⋊Dic7 [×2], C28⋊7D4 [×2], D4×C28, C2×D4⋊2D7 [×2], D4⋊6D28
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2- (1+4), D28 [×4], C22×D7 [×7], D4⋊6D4, C2×D28 [×6], D4⋊2D7 [×2], C23×D7, C22×D28, C2×D4⋊2D7, D4.10D14, D4⋊6D28
Generators and relations
G = < a,b,c,d | a4=b2=c28=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
►On 224 points
Generators in S224
(1 224 100 46)(2 197 101 47)(3 198 102 48)(4 199 103 49)(5 200 104 50)(6 201 105 51)(7 202 106 52)(8 203 107 53)(9 204 108 54)(10 205 109 55)(11 206 110 56)(12 207 111 29)(13 208 112 30)(14 209 85 31)(15 210 86 32)(16 211 87 33)(17 212 88 34)(18 213 89 35)(19 214 90 36)(20 215 91 37)(21 216 92 38)(22 217 93 39)(23 218 94 40)(24 219 95 41)(25 220 96 42)(26 221 97 43)(27 222 98 44)(28 223 99 45)(57 193 156 113)(58 194 157 114)(59 195 158 115)(60 196 159 116)(61 169 160 117)(62 170 161 118)(63 171 162 119)(64 172 163 120)(65 173 164 121)(66 174 165 122)(67 175 166 123)(68 176 167 124)(69 177 168 125)(70 178 141 126)(71 179 142 127)(72 180 143 128)(73 181 144 129)(74 182 145 130)(75 183 146 131)(76 184 147 132)(77 185 148 133)(78 186 149 134)(79 187 150 135)(80 188 151 136)(81 189 152 137)(82 190 153 138)(83 191 154 139)(84 192 155 140)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 221)(30 222)(31 223)(32 224)(33 197)(34 198)(35 199)(36 200)(37 201)(38 202)(39 203)(40 204)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 211)(48 212)(49 213)(50 214)(51 215)(52 216)(53 217)(54 218)(55 219)(56 220)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 164)(80 165)(81 166)(82 167)(83 168)(84 141)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)
(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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 123)(42 122)(43 121)(44 120)(45 119)(46 118)(47 117)(48 116)(49 115)(50 114)(51 113)(52 140)(53 139)(54 138)(55 137)(56 136)(85 148)(86 147)(87 146)(88 145)(89 144)(90 143)(91 142)(92 141)(93 168)(94 167)(95 166)(96 165)(97 164)(98 163)(99 162)(100 161)(101 160)(102 159)(103 158)(104 157)(105 156)(106 155)(107 154)(108 153)(109 152)(110 151)(111 150)(112 149)(169 197)(170 224)(171 223)(172 222)(173 221)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)(181 213)(182 212)(183 211)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)
G:=sub<Sym(224)| (1,224,100,46)(2,197,101,47)(3,198,102,48)(4,199,103,49)(5,200,104,50)(6,201,105,51)(7,202,106,52)(8,203,107,53)(9,204,108,54)(10,205,109,55)(11,206,110,56)(12,207,111,29)(13,208,112,30)(14,209,85,31)(15,210,86,32)(16,211,87,33)(17,212,88,34)(18,213,89,35)(19,214,90,36)(20,215,91,37)(21,216,92,38)(22,217,93,39)(23,218,94,40)(24,219,95,41)(25,220,96,42)(26,221,97,43)(27,222,98,44)(28,223,99,45)(57,193,156,113)(58,194,157,114)(59,195,158,115)(60,196,159,116)(61,169,160,117)(62,170,161,118)(63,171,162,119)(64,172,163,120)(65,173,164,121)(66,174,165,122)(67,175,166,123)(68,176,167,124)(69,177,168,125)(70,178,141,126)(71,179,142,127)(72,180,143,128)(73,181,144,129)(74,182,145,130)(75,183,146,131)(76,184,147,132)(77,185,148,133)(78,186,149,134)(79,187,150,135)(80,188,151,136)(81,189,152,137)(82,190,153,138)(83,191,154,139)(84,192,155,140), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,221)(30,222)(31,223)(32,224)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,141)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,123)(42,122)(43,121)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,140)(53,139)(54,138)(55,137)(56,136)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,168)(94,167)(95,166)(96,165)(97,164)(98,163)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(169,197)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)>;
G:=Group( (1,224,100,46)(2,197,101,47)(3,198,102,48)(4,199,103,49)(5,200,104,50)(6,201,105,51)(7,202,106,52)(8,203,107,53)(9,204,108,54)(10,205,109,55)(11,206,110,56)(12,207,111,29)(13,208,112,30)(14,209,85,31)(15,210,86,32)(16,211,87,33)(17,212,88,34)(18,213,89,35)(19,214,90,36)(20,215,91,37)(21,216,92,38)(22,217,93,39)(23,218,94,40)(24,219,95,41)(25,220,96,42)(26,221,97,43)(27,222,98,44)(28,223,99,45)(57,193,156,113)(58,194,157,114)(59,195,158,115)(60,196,159,116)(61,169,160,117)(62,170,161,118)(63,171,162,119)(64,172,163,120)(65,173,164,121)(66,174,165,122)(67,175,166,123)(68,176,167,124)(69,177,168,125)(70,178,141,126)(71,179,142,127)(72,180,143,128)(73,181,144,129)(74,182,145,130)(75,183,146,131)(76,184,147,132)(77,185,148,133)(78,186,149,134)(79,187,150,135)(80,188,151,136)(81,189,152,137)(82,190,153,138)(83,191,154,139)(84,192,155,140), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,221)(30,222)(31,223)(32,224)(33,197)(34,198)(35,199)(36,200)(37,201)(38,202)(39,203)(40,204)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,164)(80,165)(81,166)(82,167)(83,168)(84,141)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,126)(39,125)(40,124)(41,123)(42,122)(43,121)(44,120)(45,119)(46,118)(47,117)(48,116)(49,115)(50,114)(51,113)(52,140)(53,139)(54,138)(55,137)(56,136)(85,148)(86,147)(87,146)(88,145)(89,144)(90,143)(91,142)(92,141)(93,168)(94,167)(95,166)(96,165)(97,164)(98,163)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(169,197)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198) );
G=PermutationGroup([(1,224,100,46),(2,197,101,47),(3,198,102,48),(4,199,103,49),(5,200,104,50),(6,201,105,51),(7,202,106,52),(8,203,107,53),(9,204,108,54),(10,205,109,55),(11,206,110,56),(12,207,111,29),(13,208,112,30),(14,209,85,31),(15,210,86,32),(16,211,87,33),(17,212,88,34),(18,213,89,35),(19,214,90,36),(20,215,91,37),(21,216,92,38),(22,217,93,39),(23,218,94,40),(24,219,95,41),(25,220,96,42),(26,221,97,43),(27,222,98,44),(28,223,99,45),(57,193,156,113),(58,194,157,114),(59,195,158,115),(60,196,159,116),(61,169,160,117),(62,170,161,118),(63,171,162,119),(64,172,163,120),(65,173,164,121),(66,174,165,122),(67,175,166,123),(68,176,167,124),(69,177,168,125),(70,178,141,126),(71,179,142,127),(72,180,143,128),(73,181,144,129),(74,182,145,130),(75,183,146,131),(76,184,147,132),(77,185,148,133),(78,186,149,134),(79,187,150,135),(80,188,151,136),(81,189,152,137),(82,190,153,138),(83,191,154,139),(84,192,155,140)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,221),(30,222),(31,223),(32,224),(33,197),(34,198),(35,199),(36,200),(37,201),(38,202),(39,203),(40,204),(41,205),(42,206),(43,207),(44,208),(45,209),(46,210),(47,211),(48,212),(49,213),(50,214),(51,215),(52,216),(53,217),(54,218),(55,219),(56,220),(57,142),(58,143),(59,144),(60,145),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,164),(80,165),(81,166),(82,167),(83,168),(84,141),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196)], [(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)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,135),(30,134),(31,133),(32,132),(33,131),(34,130),(35,129),(36,128),(37,127),(38,126),(39,125),(40,124),(41,123),(42,122),(43,121),(44,120),(45,119),(46,118),(47,117),(48,116),(49,115),(50,114),(51,113),(52,140),(53,139),(54,138),(55,137),(56,136),(85,148),(86,147),(87,146),(88,145),(89,144),(90,143),(91,142),(92,141),(93,168),(94,167),(95,166),(96,165),(97,164),(98,163),(99,162),(100,161),(101,160),(102,159),(103,158),(104,157),(105,156),(106,155),(107,154),(108,153),(109,152),(110,151),(111,150),(112,149),(169,197),(170,224),(171,223),(172,222),(173,221),(174,220),(175,219),(176,218),(177,217),(178,216),(179,215),(180,214),(181,213),(182,212),(183,211),(184,210),(185,209),(186,208),(187,207),(188,206),(189,205),(190,204),(191,203),(192,202),(193,201),(194,200),(195,199),(196,198)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
| 4 | 5 | 0 | 0 |
| 22 | 6 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 3 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 12 | 17 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,28,0,0,2,1],[28,0,0,0,0,28,0,0,0,0,1,1,0,0,0,28],[4,22,0,0,5,6,0,0,0,0,28,0,0,0,0,28],[0,10,0,0,3,0,0,0,0,0,12,12,0,0,5,17] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | D14 | D28 | 2- (1+4) | D4⋊2D7 | D4.10D14 |
| kernel | D4⋊6D28 | C28⋊2Q8 | C4×D28 | C22.D28 | D14⋊2Q8 | C2×C4⋊Dic7 | C28⋊7D4 | D4×C28 | C2×D4⋊2D7 | C7×D4 | C4×D4 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 3 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_4\rtimes_6D_{28} % in TeX
G:=Group("D4:6D28"); // GroupNames label
G:=SmallGroup(448,1008);
// by ID
G=gap.SmallGroup(448,1008);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,675,80,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^28=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀