metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.30D4, C28.13C24, C56.42C23, Q16.11D14, D28.8C23, Dic14.30D4, D56.13C22, Dic14.8C23, Dic28.15C22, C7⋊2(Q8○D8), (D7×Q16)⋊6C2, C4.77(D4×D7), C7⋊C8.5C23, Q8⋊D7.C22, (C2×Q16)⋊12D7, (C14×Q16)⋊3C2, C28.88(C2×D4), C7⋊D4.10D4, Q8.D14⋊6C2, D56⋊7C2⋊4C2, Q16⋊D7⋊5C2, D14.28(C2×D4), (C2×C8).105D14, (C2×Q8).91D14, (C4×D7).7C23, (C8×D7).7C22, C8.14(C22×D7), C4.13(C23×D7), C22.22(D4×D7), D28.2C4⋊3C2, (Q8×D7).1C22, Q8.7(C22×D7), (C7×Q8).7C23, C28.C23⋊8C2, (C2×C56).35C22, Dic7.33(C2×D4), C8⋊D7.3C22, C56⋊C2.3C22, C7⋊Q16.1C22, (C2×C28).530C23, Q8.10D14⋊4C2, C4○D28.53C22, C14.114(C22×D4), (C7×Q16).11C22, Q8⋊2D7.1C22, (Q8×C14).152C22, C4.Dic7.48C22, C2.87(C2×D4×D7), (C2×C14).403(C2×D4), (C2×C4).231(C22×D7), SmallGroup(448,1219)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1156 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4 [×11], Q8 [×4], Q8 [×9], D7 [×4], C14, C14, C2×C8, C2×C8 [×2], M4(2) [×3], D8, SD16 [×6], Q16 [×4], Q16 [×5], C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×13], Dic7 [×2], Dic7 [×2], C28 [×2], C28 [×4], D14 [×2], D14 [×2], C2×C14, C8○D4, C2×Q16, C2×Q16 [×2], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14, Dic14 [×2], Dic14 [×4], C4×D7 [×2], C4×D7 [×10], D28, D28 [×2], D28 [×4], C7⋊D4 [×2], C7⋊D4 [×2], C2×C28, C2×C28 [×2], C7×Q8 [×4], C7×Q8 [×2], Q8○D8, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56, Dic28, C4.Dic7, Q8⋊D7 [×4], C7⋊Q16 [×4], C2×C56, C7×Q16 [×4], C4○D28, C4○D28 [×2], C4○D28 [×4], Q8×D7 [×4], Q8×D7 [×2], Q8⋊2D7 [×4], Q8⋊2D7 [×2], Q8×C14 [×2], D28.2C4, D56⋊7C2, D7×Q16 [×2], Q16⋊D7 [×4], Q8.D14 [×2], C28.C23 [×2], C14×Q16, Q8.10D14 [×2], D28.30D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C22×D7 [×7], Q8○D8, D4×D7 [×2], C23×D7, C2×D4×D7, D28.30D4
Generators and relations
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, ac=ca, dad-1=a15, bc=cb, dbd-1=a14b, dcd-1=a14c3 >
►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 41)(30 40)(31 39)(32 38)(33 37)(34 36)(42 56)(43 55)(44 54)(45 53)(46 52)(47 51)(48 50)(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)(85 111)(86 110)(87 109)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(128 140)(129 139)(130 138)(131 137)(132 136)(133 135)(141 147)(142 146)(143 145)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(169 173)(170 172)(174 196)(175 195)(176 194)(177 193)(178 192)(179 191)(180 190)(181 189)(182 188)(183 187)(184 186)(197 219)(198 218)(199 217)(200 216)(201 215)(202 214)(203 213)(204 212)(205 211)(206 210)(207 209)(220 224)(221 223)
(1 182 59 46 15 196 73 32)(2 183 60 47 16 169 74 33)(3 184 61 48 17 170 75 34)(4 185 62 49 18 171 76 35)(5 186 63 50 19 172 77 36)(6 187 64 51 20 173 78 37)(7 188 65 52 21 174 79 38)(8 189 66 53 22 175 80 39)(9 190 67 54 23 176 81 40)(10 191 68 55 24 177 82 41)(11 192 69 56 25 178 83 42)(12 193 70 29 26 179 84 43)(13 194 71 30 27 180 57 44)(14 195 72 31 28 181 58 45)(85 223 121 159 99 209 135 145)(86 224 122 160 100 210 136 146)(87 197 123 161 101 211 137 147)(88 198 124 162 102 212 138 148)(89 199 125 163 103 213 139 149)(90 200 126 164 104 214 140 150)(91 201 127 165 105 215 113 151)(92 202 128 166 106 216 114 152)(93 203 129 167 107 217 115 153)(94 204 130 168 108 218 116 154)(95 205 131 141 109 219 117 155)(96 206 132 142 110 220 118 156)(97 207 133 143 111 221 119 157)(98 208 134 144 112 222 120 158)
(1 102 15 88)(2 89 16 103)(3 104 17 90)(4 91 18 105)(5 106 19 92)(6 93 20 107)(7 108 21 94)(8 95 22 109)(9 110 23 96)(10 97 24 111)(11 112 25 98)(12 99 26 85)(13 86 27 100)(14 101 28 87)(29 223 43 209)(30 210 44 224)(31 197 45 211)(32 212 46 198)(33 199 47 213)(34 214 48 200)(35 201 49 215)(36 216 50 202)(37 203 51 217)(38 218 52 204)(39 205 53 219)(40 220 54 206)(41 207 55 221)(42 222 56 208)(57 122 71 136)(58 137 72 123)(59 124 73 138)(60 139 74 125)(61 126 75 140)(62 113 76 127)(63 128 77 114)(64 115 78 129)(65 130 79 116)(66 117 80 131)(67 132 81 118)(68 119 82 133)(69 134 83 120)(70 121 84 135)(141 189 155 175)(142 176 156 190)(143 191 157 177)(144 178 158 192)(145 193 159 179)(146 180 160 194)(147 195 161 181)(148 182 162 196)(149 169 163 183)(150 184 164 170)(151 171 165 185)(152 186 166 172)(153 173 167 187)(154 188 168 174)
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)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(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)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(169,173)(170,172)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(197,219)(198,218)(199,217)(200,216)(201,215)(202,214)(203,213)(204,212)(205,211)(206,210)(207,209)(220,224)(221,223), (1,182,59,46,15,196,73,32)(2,183,60,47,16,169,74,33)(3,184,61,48,17,170,75,34)(4,185,62,49,18,171,76,35)(5,186,63,50,19,172,77,36)(6,187,64,51,20,173,78,37)(7,188,65,52,21,174,79,38)(8,189,66,53,22,175,80,39)(9,190,67,54,23,176,81,40)(10,191,68,55,24,177,82,41)(11,192,69,56,25,178,83,42)(12,193,70,29,26,179,84,43)(13,194,71,30,27,180,57,44)(14,195,72,31,28,181,58,45)(85,223,121,159,99,209,135,145)(86,224,122,160,100,210,136,146)(87,197,123,161,101,211,137,147)(88,198,124,162,102,212,138,148)(89,199,125,163,103,213,139,149)(90,200,126,164,104,214,140,150)(91,201,127,165,105,215,113,151)(92,202,128,166,106,216,114,152)(93,203,129,167,107,217,115,153)(94,204,130,168,108,218,116,154)(95,205,131,141,109,219,117,155)(96,206,132,142,110,220,118,156)(97,207,133,143,111,221,119,157)(98,208,134,144,112,222,120,158), (1,102,15,88)(2,89,16,103)(3,104,17,90)(4,91,18,105)(5,106,19,92)(6,93,20,107)(7,108,21,94)(8,95,22,109)(9,110,23,96)(10,97,24,111)(11,112,25,98)(12,99,26,85)(13,86,27,100)(14,101,28,87)(29,223,43,209)(30,210,44,224)(31,197,45,211)(32,212,46,198)(33,199,47,213)(34,214,48,200)(35,201,49,215)(36,216,50,202)(37,203,51,217)(38,218,52,204)(39,205,53,219)(40,220,54,206)(41,207,55,221)(42,222,56,208)(57,122,71,136)(58,137,72,123)(59,124,73,138)(60,139,74,125)(61,126,75,140)(62,113,76,127)(63,128,77,114)(64,115,78,129)(65,130,79,116)(66,117,80,131)(67,132,81,118)(68,119,82,133)(69,134,83,120)(70,121,84,135)(141,189,155,175)(142,176,156,190)(143,191,157,177)(144,178,158,192)(145,193,159,179)(146,180,160,194)(147,195,161,181)(148,182,162,196)(149,169,163,183)(150,184,164,170)(151,171,165,185)(152,186,166,172)(153,173,167,187)(154,188,168,174)>;
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)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(42,56)(43,55)(44,54)(45,53)(46,52)(47,51)(48,50)(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)(85,111)(86,110)(87,109)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(128,140)(129,139)(130,138)(131,137)(132,136)(133,135)(141,147)(142,146)(143,145)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(169,173)(170,172)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(197,219)(198,218)(199,217)(200,216)(201,215)(202,214)(203,213)(204,212)(205,211)(206,210)(207,209)(220,224)(221,223), (1,182,59,46,15,196,73,32)(2,183,60,47,16,169,74,33)(3,184,61,48,17,170,75,34)(4,185,62,49,18,171,76,35)(5,186,63,50,19,172,77,36)(6,187,64,51,20,173,78,37)(7,188,65,52,21,174,79,38)(8,189,66,53,22,175,80,39)(9,190,67,54,23,176,81,40)(10,191,68,55,24,177,82,41)(11,192,69,56,25,178,83,42)(12,193,70,29,26,179,84,43)(13,194,71,30,27,180,57,44)(14,195,72,31,28,181,58,45)(85,223,121,159,99,209,135,145)(86,224,122,160,100,210,136,146)(87,197,123,161,101,211,137,147)(88,198,124,162,102,212,138,148)(89,199,125,163,103,213,139,149)(90,200,126,164,104,214,140,150)(91,201,127,165,105,215,113,151)(92,202,128,166,106,216,114,152)(93,203,129,167,107,217,115,153)(94,204,130,168,108,218,116,154)(95,205,131,141,109,219,117,155)(96,206,132,142,110,220,118,156)(97,207,133,143,111,221,119,157)(98,208,134,144,112,222,120,158), (1,102,15,88)(2,89,16,103)(3,104,17,90)(4,91,18,105)(5,106,19,92)(6,93,20,107)(7,108,21,94)(8,95,22,109)(9,110,23,96)(10,97,24,111)(11,112,25,98)(12,99,26,85)(13,86,27,100)(14,101,28,87)(29,223,43,209)(30,210,44,224)(31,197,45,211)(32,212,46,198)(33,199,47,213)(34,214,48,200)(35,201,49,215)(36,216,50,202)(37,203,51,217)(38,218,52,204)(39,205,53,219)(40,220,54,206)(41,207,55,221)(42,222,56,208)(57,122,71,136)(58,137,72,123)(59,124,73,138)(60,139,74,125)(61,126,75,140)(62,113,76,127)(63,128,77,114)(64,115,78,129)(65,130,79,116)(66,117,80,131)(67,132,81,118)(68,119,82,133)(69,134,83,120)(70,121,84,135)(141,189,155,175)(142,176,156,190)(143,191,157,177)(144,178,158,192)(145,193,159,179)(146,180,160,194)(147,195,161,181)(148,182,162,196)(149,169,163,183)(150,184,164,170)(151,171,165,185)(152,186,166,172)(153,173,167,187)(154,188,168,174) );
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),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(42,56),(43,55),(44,54),(45,53),(46,52),(47,51),(48,50),(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),(85,111),(86,110),(87,109),(88,108),(89,107),(90,106),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(128,140),(129,139),(130,138),(131,137),(132,136),(133,135),(141,147),(142,146),(143,145),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(169,173),(170,172),(174,196),(175,195),(176,194),(177,193),(178,192),(179,191),(180,190),(181,189),(182,188),(183,187),(184,186),(197,219),(198,218),(199,217),(200,216),(201,215),(202,214),(203,213),(204,212),(205,211),(206,210),(207,209),(220,224),(221,223)], [(1,182,59,46,15,196,73,32),(2,183,60,47,16,169,74,33),(3,184,61,48,17,170,75,34),(4,185,62,49,18,171,76,35),(5,186,63,50,19,172,77,36),(6,187,64,51,20,173,78,37),(7,188,65,52,21,174,79,38),(8,189,66,53,22,175,80,39),(9,190,67,54,23,176,81,40),(10,191,68,55,24,177,82,41),(11,192,69,56,25,178,83,42),(12,193,70,29,26,179,84,43),(13,194,71,30,27,180,57,44),(14,195,72,31,28,181,58,45),(85,223,121,159,99,209,135,145),(86,224,122,160,100,210,136,146),(87,197,123,161,101,211,137,147),(88,198,124,162,102,212,138,148),(89,199,125,163,103,213,139,149),(90,200,126,164,104,214,140,150),(91,201,127,165,105,215,113,151),(92,202,128,166,106,216,114,152),(93,203,129,167,107,217,115,153),(94,204,130,168,108,218,116,154),(95,205,131,141,109,219,117,155),(96,206,132,142,110,220,118,156),(97,207,133,143,111,221,119,157),(98,208,134,144,112,222,120,158)], [(1,102,15,88),(2,89,16,103),(3,104,17,90),(4,91,18,105),(5,106,19,92),(6,93,20,107),(7,108,21,94),(8,95,22,109),(9,110,23,96),(10,97,24,111),(11,112,25,98),(12,99,26,85),(13,86,27,100),(14,101,28,87),(29,223,43,209),(30,210,44,224),(31,197,45,211),(32,212,46,198),(33,199,47,213),(34,214,48,200),(35,201,49,215),(36,216,50,202),(37,203,51,217),(38,218,52,204),(39,205,53,219),(40,220,54,206),(41,207,55,221),(42,222,56,208),(57,122,71,136),(58,137,72,123),(59,124,73,138),(60,139,74,125),(61,126,75,140),(62,113,76,127),(63,128,77,114),(64,115,78,129),(65,130,79,116),(66,117,80,131),(67,132,81,118),(68,119,82,133),(69,134,83,120),(70,121,84,135),(141,189,155,175),(142,176,156,190),(143,191,157,177),(144,178,158,192),(145,193,159,179),(146,180,160,194),(147,195,161,181),(148,182,162,196),(149,169,163,183),(150,184,164,170),(151,171,165,185),(152,186,166,172),(153,173,167,187),(154,188,168,174)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 59 | 31 | 54 | 99 |
| 82 | 22 | 93 | 6 |
| 58 | 81 | 107 | 51 |
| 51 | 90 | 31 | 38 |
| 1 | 34 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 58 | 69 |
| 0 | 0 | 79 | 55 |
| 0 | 0 | 0 | 59 |
| 0 | 0 | 54 | 107 |
| 10 | 23 | 51 | 0 |
| 90 | 0 | 0 | 51 |
| 40 | 9 | 82 | 91 |
| 104 | 73 | 75 | 36 |
| 41 | 69 | 41 | 95 |
| 37 | 62 | 9 | 72 |
G:=sub<GL(4,GF(113))| [59,82,58,51,31,22,81,90,54,93,107,31,99,6,51,38],[1,0,0,0,34,112,0,0,0,0,58,79,0,0,69,55],[0,0,10,90,0,0,23,0,0,54,51,0,59,107,0,51],[40,104,41,37,9,73,69,62,82,75,41,9,91,36,95,72] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 14 | 14 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | Q8○D8 | D4×D7 | D4×D7 | D28.30D4 |
| kernel | D28.30D4 | D28.2C4 | D56⋊7C2 | D7×Q16 | Q16⋊D7 | Q8.D14 | C28.C23 | C14×Q16 | Q8.10D14 | Dic14 | D28 | C7⋊D4 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 3 | 12 | 6 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
D_{28}._{30}D_4 % in TeX
G:=Group("D28.30D4"); // GroupNames label
G:=SmallGroup(448,1219);
// by ID
G=gap.SmallGroup(448,1219);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,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=a^14*c^3>;
// generators/relations
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