metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊8Q8, Dic14⋊12D4, C42.173D14, C14.362- (1+4), C7⋊4(D4×Q8), C4⋊1(Q8×D7), C4⋊Q8⋊11D7, C28⋊3(C2×Q8), C28⋊Q8⋊44C2, D14⋊7(C2×Q8), C4.74(D4×D7), C28.72(C2×D4), C4⋊C4.218D14, (C4×D28).26C2, D14⋊3Q8⋊36C2, D14⋊Q8⋊48C2, (C4×Dic14)⋊52C2, (C2×Q8).146D14, Dic7.30(C2×D4), D28⋊C4.13C2, C14.47(C22×Q8), (C4×C28).212C22, (C2×C14).271C24, (C2×C28).104C23, C14.101(C22×D4), D14⋊C4.152C22, (C2×D28).272C22, C4⋊Dic7.385C22, (Q8×C14).138C22, C22.292(C23×D7), Dic7⋊C4.166C22, (C4×Dic7).160C22, (C2×Dic7).142C23, (C22×D7).232C23, C2.37(Q8.10D14), (C2×Dic14).189C22, (C2×Q8×D7)⋊13C2, C2.74(C2×D4×D7), C2.30(C2×Q8×D7), (C7×C4⋊Q8)⋊13C2, (C2×C4×D7).145C22, (C7×C4⋊C4).214C22, (C2×C4).218(C22×D7), SmallGroup(448,1180)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1260 in 280 conjugacy classes, 115 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×13], C22, C22 [×8], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×18], D4 [×4], Q8 [×16], C23 [×2], D7 [×4], C14 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8 [×13], Dic7 [×4], Dic7 [×4], C28 [×4], C28 [×5], D14 [×4], D14 [×4], C2×C14, C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8, C4⋊Q8 [×2], C22×Q8 [×2], Dic14 [×4], Dic14 [×8], C4×D7 [×12], D28 [×4], C2×Dic7 [×6], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×4], C22×D7 [×2], D4×Q8, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×6], C4×C28, C7×C4⋊C4 [×4], C2×Dic14, C2×Dic14 [×4], C2×C4×D7 [×6], C2×D28, Q8×D7 [×8], Q8×C14 [×2], C4×Dic14, C4×D28, C28⋊Q8 [×2], D28⋊C4 [×2], D14⋊Q8 [×4], D14⋊3Q8 [×2], C7×C4⋊Q8, C2×Q8×D7 [×2], D28⋊8Q8
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D7, C2×D4 [×6], C2×Q8 [×6], C24, D14 [×7], C22×D4, C22×Q8, 2- (1+4), C22×D7 [×7], D4×Q8, D4×D7 [×2], Q8×D7 [×2], C23×D7, C2×D4×D7, C2×Q8×D7, Q8.10D14, D28⋊8Q8
Generators and relations
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a15, bc=cb, dbd-1=a14b, dcd-1=c-1 >
►On 224 points
Generators in S224
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 67)(58 66)(59 65)(60 64)(61 63)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(102 112)(103 111)(104 110)(105 109)(106 108)(113 115)(116 140)(117 139)(118 138)(119 137)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(142 168)(143 167)(144 166)(145 165)(146 164)(147 163)(148 162)(149 161)(150 160)(151 159)(152 158)(153 157)(154 156)(169 195)(170 194)(171 193)(172 192)(173 191)(174 190)(175 189)(176 188)(177 187)(178 186)(179 185)(180 184)(181 183)(197 205)(198 204)(199 203)(200 202)(206 224)(207 223)(208 222)(209 221)(210 220)(211 219)(212 218)(213 217)(214 216)
(1 73 33 139)(2 74 34 140)(3 75 35 113)(4 76 36 114)(5 77 37 115)(6 78 38 116)(7 79 39 117)(8 80 40 118)(9 81 41 119)(10 82 42 120)(11 83 43 121)(12 84 44 122)(13 57 45 123)(14 58 46 124)(15 59 47 125)(16 60 48 126)(17 61 49 127)(18 62 50 128)(19 63 51 129)(20 64 52 130)(21 65 53 131)(22 66 54 132)(23 67 55 133)(24 68 56 134)(25 69 29 135)(26 70 30 136)(27 71 31 137)(28 72 32 138)(85 161 221 174)(86 162 222 175)(87 163 223 176)(88 164 224 177)(89 165 197 178)(90 166 198 179)(91 167 199 180)(92 168 200 181)(93 141 201 182)(94 142 202 183)(95 143 203 184)(96 144 204 185)(97 145 205 186)(98 146 206 187)(99 147 207 188)(100 148 208 189)(101 149 209 190)(102 150 210 191)(103 151 211 192)(104 152 212 193)(105 153 213 194)(106 154 214 195)(107 155 215 196)(108 156 216 169)(109 157 217 170)(110 158 218 171)(111 159 219 172)(112 160 220 173)
(1 186 33 145)(2 173 34 160)(3 188 35 147)(4 175 36 162)(5 190 37 149)(6 177 38 164)(7 192 39 151)(8 179 40 166)(9 194 41 153)(10 181 42 168)(11 196 43 155)(12 183 44 142)(13 170 45 157)(14 185 46 144)(15 172 47 159)(16 187 48 146)(17 174 49 161)(18 189 50 148)(19 176 51 163)(20 191 52 150)(21 178 53 165)(22 193 54 152)(23 180 55 167)(24 195 56 154)(25 182 29 141)(26 169 30 156)(27 184 31 143)(28 171 32 158)(57 217 123 109)(58 204 124 96)(59 219 125 111)(60 206 126 98)(61 221 127 85)(62 208 128 100)(63 223 129 87)(64 210 130 102)(65 197 131 89)(66 212 132 104)(67 199 133 91)(68 214 134 106)(69 201 135 93)(70 216 136 108)(71 203 137 95)(72 218 138 110)(73 205 139 97)(74 220 140 112)(75 207 113 99)(76 222 114 86)(77 209 115 101)(78 224 116 88)(79 211 117 103)(80 198 118 90)(81 213 119 105)(82 200 120 92)(83 215 121 107)(84 202 122 94)
G:=sub<Sym(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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(102,112)(103,111)(104,110)(105,109)(106,108)(113,115)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(142,168)(143,167)(144,166)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)(169,195)(170,194)(171,193)(172,192)(173,191)(174,190)(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(181,183)(197,205)(198,204)(199,203)(200,202)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216), (1,73,33,139)(2,74,34,140)(3,75,35,113)(4,76,36,114)(5,77,37,115)(6,78,38,116)(7,79,39,117)(8,80,40,118)(9,81,41,119)(10,82,42,120)(11,83,43,121)(12,84,44,122)(13,57,45,123)(14,58,46,124)(15,59,47,125)(16,60,48,126)(17,61,49,127)(18,62,50,128)(19,63,51,129)(20,64,52,130)(21,65,53,131)(22,66,54,132)(23,67,55,133)(24,68,56,134)(25,69,29,135)(26,70,30,136)(27,71,31,137)(28,72,32,138)(85,161,221,174)(86,162,222,175)(87,163,223,176)(88,164,224,177)(89,165,197,178)(90,166,198,179)(91,167,199,180)(92,168,200,181)(93,141,201,182)(94,142,202,183)(95,143,203,184)(96,144,204,185)(97,145,205,186)(98,146,206,187)(99,147,207,188)(100,148,208,189)(101,149,209,190)(102,150,210,191)(103,151,211,192)(104,152,212,193)(105,153,213,194)(106,154,214,195)(107,155,215,196)(108,156,216,169)(109,157,217,170)(110,158,218,171)(111,159,219,172)(112,160,220,173), (1,186,33,145)(2,173,34,160)(3,188,35,147)(4,175,36,162)(5,190,37,149)(6,177,38,164)(7,192,39,151)(8,179,40,166)(9,194,41,153)(10,181,42,168)(11,196,43,155)(12,183,44,142)(13,170,45,157)(14,185,46,144)(15,172,47,159)(16,187,48,146)(17,174,49,161)(18,189,50,148)(19,176,51,163)(20,191,52,150)(21,178,53,165)(22,193,54,152)(23,180,55,167)(24,195,56,154)(25,182,29,141)(26,169,30,156)(27,184,31,143)(28,171,32,158)(57,217,123,109)(58,204,124,96)(59,219,125,111)(60,206,126,98)(61,221,127,85)(62,208,128,100)(63,223,129,87)(64,210,130,102)(65,197,131,89)(66,212,132,104)(67,199,133,91)(68,214,134,106)(69,201,135,93)(70,216,136,108)(71,203,137,95)(72,218,138,110)(73,205,139,97)(74,220,140,112)(75,207,113,99)(76,222,114,86)(77,209,115,101)(78,224,116,88)(79,211,117,103)(80,198,118,90)(81,213,119,105)(82,200,120,92)(83,215,121,107)(84,202,122,94)>;
G:=Group( (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,43)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,67)(58,66)(59,65)(60,64)(61,63)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(85,101)(86,100)(87,99)(88,98)(89,97)(90,96)(91,95)(92,94)(102,112)(103,111)(104,110)(105,109)(106,108)(113,115)(116,140)(117,139)(118,138)(119,137)(120,136)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(142,168)(143,167)(144,166)(145,165)(146,164)(147,163)(148,162)(149,161)(150,160)(151,159)(152,158)(153,157)(154,156)(169,195)(170,194)(171,193)(172,192)(173,191)(174,190)(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(181,183)(197,205)(198,204)(199,203)(200,202)(206,224)(207,223)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,216), (1,73,33,139)(2,74,34,140)(3,75,35,113)(4,76,36,114)(5,77,37,115)(6,78,38,116)(7,79,39,117)(8,80,40,118)(9,81,41,119)(10,82,42,120)(11,83,43,121)(12,84,44,122)(13,57,45,123)(14,58,46,124)(15,59,47,125)(16,60,48,126)(17,61,49,127)(18,62,50,128)(19,63,51,129)(20,64,52,130)(21,65,53,131)(22,66,54,132)(23,67,55,133)(24,68,56,134)(25,69,29,135)(26,70,30,136)(27,71,31,137)(28,72,32,138)(85,161,221,174)(86,162,222,175)(87,163,223,176)(88,164,224,177)(89,165,197,178)(90,166,198,179)(91,167,199,180)(92,168,200,181)(93,141,201,182)(94,142,202,183)(95,143,203,184)(96,144,204,185)(97,145,205,186)(98,146,206,187)(99,147,207,188)(100,148,208,189)(101,149,209,190)(102,150,210,191)(103,151,211,192)(104,152,212,193)(105,153,213,194)(106,154,214,195)(107,155,215,196)(108,156,216,169)(109,157,217,170)(110,158,218,171)(111,159,219,172)(112,160,220,173), (1,186,33,145)(2,173,34,160)(3,188,35,147)(4,175,36,162)(5,190,37,149)(6,177,38,164)(7,192,39,151)(8,179,40,166)(9,194,41,153)(10,181,42,168)(11,196,43,155)(12,183,44,142)(13,170,45,157)(14,185,46,144)(15,172,47,159)(16,187,48,146)(17,174,49,161)(18,189,50,148)(19,176,51,163)(20,191,52,150)(21,178,53,165)(22,193,54,152)(23,180,55,167)(24,195,56,154)(25,182,29,141)(26,169,30,156)(27,184,31,143)(28,171,32,158)(57,217,123,109)(58,204,124,96)(59,219,125,111)(60,206,126,98)(61,221,127,85)(62,208,128,100)(63,223,129,87)(64,210,130,102)(65,197,131,89)(66,212,132,104)(67,199,133,91)(68,214,134,106)(69,201,135,93)(70,216,136,108)(71,203,137,95)(72,218,138,110)(73,205,139,97)(74,220,140,112)(75,207,113,99)(76,222,114,86)(77,209,115,101)(78,224,116,88)(79,211,117,103)(80,198,118,90)(81,213,119,105)(82,200,120,92)(83,215,121,107)(84,202,122,94) );
G=PermutationGroup([(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,43),(30,42),(31,41),(32,40),(33,39),(34,38),(35,37),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(57,67),(58,66),(59,65),(60,64),(61,63),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(85,101),(86,100),(87,99),(88,98),(89,97),(90,96),(91,95),(92,94),(102,112),(103,111),(104,110),(105,109),(106,108),(113,115),(116,140),(117,139),(118,138),(119,137),(120,136),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(142,168),(143,167),(144,166),(145,165),(146,164),(147,163),(148,162),(149,161),(150,160),(151,159),(152,158),(153,157),(154,156),(169,195),(170,194),(171,193),(172,192),(173,191),(174,190),(175,189),(176,188),(177,187),(178,186),(179,185),(180,184),(181,183),(197,205),(198,204),(199,203),(200,202),(206,224),(207,223),(208,222),(209,221),(210,220),(211,219),(212,218),(213,217),(214,216)], [(1,73,33,139),(2,74,34,140),(3,75,35,113),(4,76,36,114),(5,77,37,115),(6,78,38,116),(7,79,39,117),(8,80,40,118),(9,81,41,119),(10,82,42,120),(11,83,43,121),(12,84,44,122),(13,57,45,123),(14,58,46,124),(15,59,47,125),(16,60,48,126),(17,61,49,127),(18,62,50,128),(19,63,51,129),(20,64,52,130),(21,65,53,131),(22,66,54,132),(23,67,55,133),(24,68,56,134),(25,69,29,135),(26,70,30,136),(27,71,31,137),(28,72,32,138),(85,161,221,174),(86,162,222,175),(87,163,223,176),(88,164,224,177),(89,165,197,178),(90,166,198,179),(91,167,199,180),(92,168,200,181),(93,141,201,182),(94,142,202,183),(95,143,203,184),(96,144,204,185),(97,145,205,186),(98,146,206,187),(99,147,207,188),(100,148,208,189),(101,149,209,190),(102,150,210,191),(103,151,211,192),(104,152,212,193),(105,153,213,194),(106,154,214,195),(107,155,215,196),(108,156,216,169),(109,157,217,170),(110,158,218,171),(111,159,219,172),(112,160,220,173)], [(1,186,33,145),(2,173,34,160),(3,188,35,147),(4,175,36,162),(5,190,37,149),(6,177,38,164),(7,192,39,151),(8,179,40,166),(9,194,41,153),(10,181,42,168),(11,196,43,155),(12,183,44,142),(13,170,45,157),(14,185,46,144),(15,172,47,159),(16,187,48,146),(17,174,49,161),(18,189,50,148),(19,176,51,163),(20,191,52,150),(21,178,53,165),(22,193,54,152),(23,180,55,167),(24,195,56,154),(25,182,29,141),(26,169,30,156),(27,184,31,143),(28,171,32,158),(57,217,123,109),(58,204,124,96),(59,219,125,111),(60,206,126,98),(61,221,127,85),(62,208,128,100),(63,223,129,87),(64,210,130,102),(65,197,131,89),(66,212,132,104),(67,199,133,91),(68,214,134,106),(69,201,135,93),(70,216,136,108),(71,203,137,95),(72,218,138,110),(73,205,139,97),(74,220,140,112),(75,207,113,99),(76,222,114,86),(77,209,115,101),(78,224,116,88),(79,211,117,103),(80,198,118,90),(81,213,119,105),(82,200,120,92),(83,215,121,107),(84,202,122,94)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 15 | 18 | 0 | 0 | 0 | 0 |
| 22 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 4 | 0 | 0 | 0 | 0 |
| 15 | 17 | 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 |
| 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 | 23 | 5 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 26 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [15,22,0,0,0,0,18,11,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,15,0,0,0,0,4,17,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,23,10,0,0,0,0,5,6],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,26,17] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | D7 | D14 | D14 | D14 | 2- (1+4) | D4×D7 | Q8×D7 | Q8.10D14 |
| kernel | D28⋊8Q8 | C4×Dic14 | C4×D28 | C28⋊Q8 | D28⋊C4 | D14⋊Q8 | D14⋊3Q8 | C7×C4⋊Q8 | C2×Q8×D7 | Dic14 | D28 | C4⋊Q8 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 3 | 3 | 12 | 6 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_8Q_8 % in TeX
G:=Group("D28:8Q8"); // GroupNames label
G:=SmallGroup(448,1180);
// by ID
G=gap.SmallGroup(448,1180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,1571,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^15,b*c=c*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations
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