metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.44D4, C56.7C23, Q16.6D14, C28.26C24, SD16.1D14, Dic14.44D4, D28.19C23, M4(2).18D14, Dic28.1C22, Dic14.19C23, C7⋊4(Q8○D8), (D7×Q16)⋊2C2, C7⋊D4.7D4, D4⋊D7.C22, D28.C4⋊4C2, C4.118(D4×D7), C7⋊C8.28C23, C8⋊D7.C22, C8.C22⋊5D7, C56⋊C2.C22, C8.7(C22×D7), Q16⋊D7⋊4C2, C4○D4.15D14, D14.35(C2×D4), C8.D14⋊4C2, SD16⋊D7⋊4C2, C28.247(C2×D4), (C2×Q8).92D14, (C8×D7).2C22, C22.17(D4×D7), C4.26(C23×D7), Q8⋊D7.2C22, (C7×SD16).C22, SD16⋊3D7⋊4C2, D4.8D14⋊6C2, (Q8×D7).3C22, Dic7.40(C2×D4), (C7×D4).19C23, (C4×D7).17C23, D4.19(C22×D7), D4.D7.2C22, D4.10D14⋊8C2, (C7×Q8).19C23, Q8.19(C22×D7), (C7×Q16).1C22, C7⋊Q16.3C22, (C2×C28).117C23, Q8.10D14⋊6C2, C4○D28.32C22, D4⋊2D7.3C22, C14.127(C22×D4), Q8⋊2D7.3C22, (Q8×C14).153C22, (C7×M4(2)).1C22, (C2×Dic14).201C22, C2.100(C2×D4×D7), (C2×C7⋊Q16)⋊29C2, (C2×C14).72(C2×D4), (C7×C8.C22)⋊4C2, (C2×C7⋊C8).182C22, (C7×C4○D4).28C22, (C2×C4).101(C22×D7), SmallGroup(448,1232)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1132 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4, D4 [×10], Q8, Q8 [×2], Q8 [×10], D7 [×3], C14, C14 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8, SD16 [×2], SD16 [×4], Q16 [×2], Q16 [×7], C2×Q8, C2×Q8 [×7], C4○D4, C4○D4 [×12], Dic7 [×2], Dic7 [×3], C28 [×2], C28 [×3], D14 [×2], D14, C2×C14, C2×C14, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22, C8.C22 [×5], 2- (1+4) [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×2], Dic14 [×2], Dic14 [×5], C4×D7 [×2], C4×D7 [×7], D28 [×2], D28 [×2], C2×Dic7 [×3], C7⋊D4 [×2], C7⋊D4 [×3], C2×C28, C2×C28 [×2], C7×D4, C7×D4, C7×Q8, C7×Q8 [×2], C7×Q8, Q8○D8, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], Dic28 [×2], C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C7⋊Q16 [×4], C7×M4(2), C7×SD16 [×2], C7×Q16 [×2], C2×Dic14, C2×Dic14, C4○D28 [×2], C4○D28 [×3], D4⋊2D7 [×2], D4⋊2D7 [×2], Q8×D7 [×4], Q8×D7, Q8⋊2D7 [×2], Q8⋊2D7, Q8×C14, C7×C4○D4, D28.C4, C8.D14, SD16⋊D7 [×2], SD16⋊3D7 [×2], D7×Q16 [×2], Q16⋊D7 [×2], C2×C7⋊Q16, D4.8D14, C7×C8.C22, Q8.10D14, D4.10D14, D28.44D4
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, D28.44D4
Generators and relations
G = < a,b,c,d | a28=b2=1, c4=d2=a14, bab=a-1, cac-1=dad-1=a15, cbc-1=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)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(85 107)(86 106)(87 105)(88 104)(89 103)(90 102)(91 101)(92 100)(93 99)(94 98)(95 97)(108 112)(109 111)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(136 140)(137 139)(141 167)(142 166)(143 165)(144 164)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)(169 185)(170 184)(171 183)(172 182)(173 181)(174 180)(175 179)(176 178)(186 196)(187 195)(188 194)(189 193)(190 192)(197 209)(198 208)(199 207)(200 206)(201 205)(202 204)(210 224)(211 223)(212 222)(213 221)(214 220)(215 219)(216 218)
(1 86 188 82 15 100 174 68)(2 101 189 69 16 87 175 83)(3 88 190 84 17 102 176 70)(4 103 191 71 18 89 177 57)(5 90 192 58 19 104 178 72)(6 105 193 73 20 91 179 59)(7 92 194 60 21 106 180 74)(8 107 195 75 22 93 181 61)(9 94 196 62 23 108 182 76)(10 109 169 77 24 95 183 63)(11 96 170 64 25 110 184 78)(12 111 171 79 26 97 185 65)(13 98 172 66 27 112 186 80)(14 85 173 81 28 99 187 67)(29 147 124 210 43 161 138 224)(30 162 125 197 44 148 139 211)(31 149 126 212 45 163 140 198)(32 164 127 199 46 150 113 213)(33 151 128 214 47 165 114 200)(34 166 129 201 48 152 115 215)(35 153 130 216 49 167 116 202)(36 168 131 203 50 154 117 217)(37 155 132 218 51 141 118 204)(38 142 133 205 52 156 119 219)(39 157 134 220 53 143 120 206)(40 144 135 207 54 158 121 221)(41 159 136 222 55 145 122 208)(42 146 137 209 56 160 123 223)
(1 158 15 144)(2 145 16 159)(3 160 17 146)(4 147 18 161)(5 162 19 148)(6 149 20 163)(7 164 21 150)(8 151 22 165)(9 166 23 152)(10 153 24 167)(11 168 25 154)(12 155 26 141)(13 142 27 156)(14 157 28 143)(29 89 43 103)(30 104 44 90)(31 91 45 105)(32 106 46 92)(33 93 47 107)(34 108 48 94)(35 95 49 109)(36 110 50 96)(37 97 51 111)(38 112 52 98)(39 99 53 85)(40 86 54 100)(41 101 55 87)(42 88 56 102)(57 124 71 138)(58 139 72 125)(59 126 73 140)(60 113 74 127)(61 128 75 114)(62 115 76 129)(63 130 77 116)(64 117 78 131)(65 132 79 118)(66 119 80 133)(67 134 81 120)(68 121 82 135)(69 136 83 122)(70 123 84 137)(169 202 183 216)(170 217 184 203)(171 204 185 218)(172 219 186 205)(173 206 187 220)(174 221 188 207)(175 208 189 222)(176 223 190 209)(177 210 191 224)(178 197 192 211)(179 212 193 198)(180 199 194 213)(181 214 195 200)(182 201 196 215)
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)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(136,140)(137,139)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(169,185)(170,184)(171,183)(172,182)(173,181)(174,180)(175,179)(176,178)(186,196)(187,195)(188,194)(189,193)(190,192)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,86,188,82,15,100,174,68)(2,101,189,69,16,87,175,83)(3,88,190,84,17,102,176,70)(4,103,191,71,18,89,177,57)(5,90,192,58,19,104,178,72)(6,105,193,73,20,91,179,59)(7,92,194,60,21,106,180,74)(8,107,195,75,22,93,181,61)(9,94,196,62,23,108,182,76)(10,109,169,77,24,95,183,63)(11,96,170,64,25,110,184,78)(12,111,171,79,26,97,185,65)(13,98,172,66,27,112,186,80)(14,85,173,81,28,99,187,67)(29,147,124,210,43,161,138,224)(30,162,125,197,44,148,139,211)(31,149,126,212,45,163,140,198)(32,164,127,199,46,150,113,213)(33,151,128,214,47,165,114,200)(34,166,129,201,48,152,115,215)(35,153,130,216,49,167,116,202)(36,168,131,203,50,154,117,217)(37,155,132,218,51,141,118,204)(38,142,133,205,52,156,119,219)(39,157,134,220,53,143,120,206)(40,144,135,207,54,158,121,221)(41,159,136,222,55,145,122,208)(42,146,137,209,56,160,123,223), (1,158,15,144)(2,145,16,159)(3,160,17,146)(4,147,18,161)(5,162,19,148)(6,149,20,163)(7,164,21,150)(8,151,22,165)(9,166,23,152)(10,153,24,167)(11,168,25,154)(12,155,26,141)(13,142,27,156)(14,157,28,143)(29,89,43,103)(30,104,44,90)(31,91,45,105)(32,106,46,92)(33,93,47,107)(34,108,48,94)(35,95,49,109)(36,110,50,96)(37,97,51,111)(38,112,52,98)(39,99,53,85)(40,86,54,100)(41,101,55,87)(42,88,56,102)(57,124,71,138)(58,139,72,125)(59,126,73,140)(60,113,74,127)(61,128,75,114)(62,115,76,129)(63,130,77,116)(64,117,78,131)(65,132,79,118)(66,119,80,133)(67,134,81,120)(68,121,82,135)(69,136,83,122)(70,123,84,137)(169,202,183,216)(170,217,184,203)(171,204,185,218)(172,219,186,205)(173,206,187,220)(174,221,188,207)(175,208,189,222)(176,223,190,209)(177,210,191,224)(178,197,192,211)(179,212,193,198)(180,199,194,213)(181,214,195,200)(182,201,196,215)>;
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)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(85,107)(86,106)(87,105)(88,104)(89,103)(90,102)(91,101)(92,100)(93,99)(94,98)(95,97)(108,112)(109,111)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(136,140)(137,139)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(169,185)(170,184)(171,183)(172,182)(173,181)(174,180)(175,179)(176,178)(186,196)(187,195)(188,194)(189,193)(190,192)(197,209)(198,208)(199,207)(200,206)(201,205)(202,204)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,86,188,82,15,100,174,68)(2,101,189,69,16,87,175,83)(3,88,190,84,17,102,176,70)(4,103,191,71,18,89,177,57)(5,90,192,58,19,104,178,72)(6,105,193,73,20,91,179,59)(7,92,194,60,21,106,180,74)(8,107,195,75,22,93,181,61)(9,94,196,62,23,108,182,76)(10,109,169,77,24,95,183,63)(11,96,170,64,25,110,184,78)(12,111,171,79,26,97,185,65)(13,98,172,66,27,112,186,80)(14,85,173,81,28,99,187,67)(29,147,124,210,43,161,138,224)(30,162,125,197,44,148,139,211)(31,149,126,212,45,163,140,198)(32,164,127,199,46,150,113,213)(33,151,128,214,47,165,114,200)(34,166,129,201,48,152,115,215)(35,153,130,216,49,167,116,202)(36,168,131,203,50,154,117,217)(37,155,132,218,51,141,118,204)(38,142,133,205,52,156,119,219)(39,157,134,220,53,143,120,206)(40,144,135,207,54,158,121,221)(41,159,136,222,55,145,122,208)(42,146,137,209,56,160,123,223), (1,158,15,144)(2,145,16,159)(3,160,17,146)(4,147,18,161)(5,162,19,148)(6,149,20,163)(7,164,21,150)(8,151,22,165)(9,166,23,152)(10,153,24,167)(11,168,25,154)(12,155,26,141)(13,142,27,156)(14,157,28,143)(29,89,43,103)(30,104,44,90)(31,91,45,105)(32,106,46,92)(33,93,47,107)(34,108,48,94)(35,95,49,109)(36,110,50,96)(37,97,51,111)(38,112,52,98)(39,99,53,85)(40,86,54,100)(41,101,55,87)(42,88,56,102)(57,124,71,138)(58,139,72,125)(59,126,73,140)(60,113,74,127)(61,128,75,114)(62,115,76,129)(63,130,77,116)(64,117,78,131)(65,132,79,118)(66,119,80,133)(67,134,81,120)(68,121,82,135)(69,136,83,122)(70,123,84,137)(169,202,183,216)(170,217,184,203)(171,204,185,218)(172,219,186,205)(173,206,187,220)(174,221,188,207)(175,208,189,222)(176,223,190,209)(177,210,191,224)(178,197,192,211)(179,212,193,198)(180,199,194,213)(181,214,195,200)(182,201,196,215) );
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),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(85,107),(86,106),(87,105),(88,104),(89,103),(90,102),(91,101),(92,100),(93,99),(94,98),(95,97),(108,112),(109,111),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(136,140),(137,139),(141,167),(142,166),(143,165),(144,164),(145,163),(146,162),(147,161),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155),(169,185),(170,184),(171,183),(172,182),(173,181),(174,180),(175,179),(176,178),(186,196),(187,195),(188,194),(189,193),(190,192),(197,209),(198,208),(199,207),(200,206),(201,205),(202,204),(210,224),(211,223),(212,222),(213,221),(214,220),(215,219),(216,218)], [(1,86,188,82,15,100,174,68),(2,101,189,69,16,87,175,83),(3,88,190,84,17,102,176,70),(4,103,191,71,18,89,177,57),(5,90,192,58,19,104,178,72),(6,105,193,73,20,91,179,59),(7,92,194,60,21,106,180,74),(8,107,195,75,22,93,181,61),(9,94,196,62,23,108,182,76),(10,109,169,77,24,95,183,63),(11,96,170,64,25,110,184,78),(12,111,171,79,26,97,185,65),(13,98,172,66,27,112,186,80),(14,85,173,81,28,99,187,67),(29,147,124,210,43,161,138,224),(30,162,125,197,44,148,139,211),(31,149,126,212,45,163,140,198),(32,164,127,199,46,150,113,213),(33,151,128,214,47,165,114,200),(34,166,129,201,48,152,115,215),(35,153,130,216,49,167,116,202),(36,168,131,203,50,154,117,217),(37,155,132,218,51,141,118,204),(38,142,133,205,52,156,119,219),(39,157,134,220,53,143,120,206),(40,144,135,207,54,158,121,221),(41,159,136,222,55,145,122,208),(42,146,137,209,56,160,123,223)], [(1,158,15,144),(2,145,16,159),(3,160,17,146),(4,147,18,161),(5,162,19,148),(6,149,20,163),(7,164,21,150),(8,151,22,165),(9,166,23,152),(10,153,24,167),(11,168,25,154),(12,155,26,141),(13,142,27,156),(14,157,28,143),(29,89,43,103),(30,104,44,90),(31,91,45,105),(32,106,46,92),(33,93,47,107),(34,108,48,94),(35,95,49,109),(36,110,50,96),(37,97,51,111),(38,112,52,98),(39,99,53,85),(40,86,54,100),(41,101,55,87),(42,88,56,102),(57,124,71,138),(58,139,72,125),(59,126,73,140),(60,113,74,127),(61,128,75,114),(62,115,76,129),(63,130,77,116),(64,117,78,131),(65,132,79,118),(66,119,80,133),(67,134,81,120),(68,121,82,135),(69,136,83,122),(70,123,84,137),(169,202,183,216),(170,217,184,203),(171,204,185,218),(172,219,186,205),(173,206,187,220),(174,221,188,207),(175,208,189,222),(176,223,190,209),(177,210,191,224),(178,197,192,211),(179,212,193,198),(180,199,194,213),(181,214,195,200),(182,201,196,215)])
Matrix representation ►G ⊆ GL8(𝔽113)
| 10 | 89 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 112 | 11 | 60 |
| 0 | 0 | 0 | 0 | 63 | 36 | 88 | 100 |
| 0 | 0 | 0 | 0 | 34 | 46 | 41 | 49 |
| 0 | 0 | 0 | 0 | 112 | 76 | 53 | 91 |
| 10 | 89 | 0 | 0 | 0 | 0 | 0 | 0 |
| 103 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 103 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 85 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 105 | 64 | 0 | 1 |
| 87 | 0 | 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 98 | 0 | 0 | 0 | 0 |
| 15 | 0 | 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 91 | 52 | 102 | 44 |
| 0 | 0 | 0 | 0 | 76 | 77 | 7 | 13 |
| 0 | 0 | 0 | 0 | 3 | 103 | 72 | 51 |
| 0 | 0 | 0 | 0 | 54 | 19 | 71 | 99 |
| 15 | 0 | 26 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 26 | 0 | 0 | 0 | 0 |
| 87 | 0 | 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 87 | 0 | 98 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 25 | 26 | 0 | 22 |
| 0 | 0 | 0 | 0 | 41 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 9 | 36 | 31 | 87 |
G:=sub<GL(8,GF(113))| [10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,10,24,0,0,0,0,0,0,89,112,0,0,0,0,0,0,0,0,58,63,34,112,0,0,0,0,112,36,46,76,0,0,0,0,11,88,41,53,0,0,0,0,60,100,49,91],[10,103,0,0,0,0,0,0,89,103,0,0,0,0,0,0,0,0,10,103,0,0,0,0,0,0,89,103,0,0,0,0,0,0,0,0,112,0,85,105,0,0,0,0,0,112,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[87,0,15,0,0,0,0,0,0,87,0,15,0,0,0,0,98,0,26,0,0,0,0,0,0,98,0,26,0,0,0,0,0,0,0,0,91,76,3,54,0,0,0,0,52,77,103,19,0,0,0,0,102,7,72,71,0,0,0,0,44,13,51,99],[15,0,87,0,0,0,0,0,0,15,0,87,0,0,0,0,26,0,98,0,0,0,0,0,0,26,0,98,0,0,0,0,0,0,0,0,61,25,41,9,0,0,0,0,0,26,0,36,0,0,0,0,36,0,52,31,0,0,0,0,0,22,0,87] >;
55 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 | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56F |
| 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 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | D14 | Q8○D8 | D4×D7 | D4×D7 | D28.44D4 |
| kernel | D28.44D4 | D28.C4 | C8.D14 | SD16⋊D7 | SD16⋊3D7 | D7×Q16 | Q16⋊D7 | C2×C7⋊Q16 | D4.8D14 | C7×C8.C22 | Q8.10D14 | D4.10D14 | Dic14 | D28 | C7⋊D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 2 | 3 | 3 | 3 |
In GAP, Magma, Sage, TeX
D_{28}._{44}D_4 % in TeX
G:=Group("D28.44D4"); // GroupNames label
G:=SmallGroup(448,1232);
// by ID
G=gap.SmallGroup(448,1232);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=d^2=a^14,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^15,c*b*c^-1=d*b*d^-1=a^14*b,d*c*d^-1=a^14*c^3>;
// generators/relations
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