metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊5Q8, C28.18SD16, C42.79D14, C4⋊Q8⋊2D7, C4.11(Q8×D7), C4⋊C4.82D14, C7⋊5(D4⋊2Q8), C28⋊C8⋊34C2, C28.38(C2×Q8), (C4×D28).18C2, (C2×C28).155D4, C4.10(Q8⋊D7), C28.81(C4○D4), C4.Dic14⋊42C2, C14.76(C2×SD16), C14.D8.14C2, C14.98(C8⋊C22), (C4×C28).131C22, (C2×C28).402C23, C4.34(Q8⋊2D7), C14.75(C22⋊Q8), C2.12(D14⋊3Q8), (C2×D28).248C22, C4⋊Dic7.347C22, C2.19(D4.D14), (C7×C4⋊Q8)⋊2C2, C2.14(C2×Q8⋊D7), (C2×C14).533(C2×D4), (C2×C7⋊C8).136C22, (C2×C4).188(C7⋊D4), (C7×C4⋊C4).129C22, (C2×C4).499(C22×D7), C22.205(C2×C7⋊D4), SmallGroup(448,618)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28⋊5Q8
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, cac-1=a15, ad=da, cbc-1=a7b, dbd-1=a14b, dcd-1=c-1 >
Subgroups: 556 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, D14, C2×C14, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C7⋊C8, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, D4⋊2Q8, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, Q8×C14, C28⋊C8, C4.Dic14, C14.D8, C4×D28, C7×C4⋊Q8, D28⋊5Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, D4⋊2Q8, Q8⋊D7, Q8×D7, Q8⋊2D7, C2×C7⋊D4, D4.D14, C2×Q8⋊D7, D14⋊3Q8, D28⋊5Q8
►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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(104 112)(105 111)(106 110)(107 109)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(133 140)(134 139)(135 138)(136 137)(141 150)(142 149)(143 148)(144 147)(145 146)(151 168)(152 167)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(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)
(1 221 137 112)(2 208 138 99)(3 223 139 86)(4 210 140 101)(5 197 113 88)(6 212 114 103)(7 199 115 90)(8 214 116 105)(9 201 117 92)(10 216 118 107)(11 203 119 94)(12 218 120 109)(13 205 121 96)(14 220 122 111)(15 207 123 98)(16 222 124 85)(17 209 125 100)(18 224 126 87)(19 211 127 102)(20 198 128 89)(21 213 129 104)(22 200 130 91)(23 215 131 106)(24 202 132 93)(25 217 133 108)(26 204 134 95)(27 219 135 110)(28 206 136 97)(29 157 58 173)(30 144 59 188)(31 159 60 175)(32 146 61 190)(33 161 62 177)(34 148 63 192)(35 163 64 179)(36 150 65 194)(37 165 66 181)(38 152 67 196)(39 167 68 183)(40 154 69 170)(41 141 70 185)(42 156 71 172)(43 143 72 187)(44 158 73 174)(45 145 74 189)(46 160 75 176)(47 147 76 191)(48 162 77 178)(49 149 78 193)(50 164 79 180)(51 151 80 195)(52 166 81 182)(53 153 82 169)(54 168 83 184)(55 155 84 171)(56 142 57 186)
(1 167 137 183)(2 168 138 184)(3 141 139 185)(4 142 140 186)(5 143 113 187)(6 144 114 188)(7 145 115 189)(8 146 116 190)(9 147 117 191)(10 148 118 192)(11 149 119 193)(12 150 120 194)(13 151 121 195)(14 152 122 196)(15 153 123 169)(16 154 124 170)(17 155 125 171)(18 156 126 172)(19 157 127 173)(20 158 128 174)(21 159 129 175)(22 160 130 176)(23 161 131 177)(24 162 132 178)(25 163 133 179)(26 164 134 180)(27 165 135 181)(28 166 136 182)(29 102 58 211)(30 103 59 212)(31 104 60 213)(32 105 61 214)(33 106 62 215)(34 107 63 216)(35 108 64 217)(36 109 65 218)(37 110 66 219)(38 111 67 220)(39 112 68 221)(40 85 69 222)(41 86 70 223)(42 87 71 224)(43 88 72 197)(44 89 73 198)(45 90 74 199)(46 91 75 200)(47 92 76 201)(48 93 77 202)(49 94 78 203)(50 95 79 204)(51 96 80 205)(52 97 81 206)(53 98 82 207)(54 99 83 208)(55 100 84 209)(56 101 57 210)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(133,140)(134,139)(135,138)(136,137)(141,150)(142,149)(143,148)(144,147)(145,146)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(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), (1,221,137,112)(2,208,138,99)(3,223,139,86)(4,210,140,101)(5,197,113,88)(6,212,114,103)(7,199,115,90)(8,214,116,105)(9,201,117,92)(10,216,118,107)(11,203,119,94)(12,218,120,109)(13,205,121,96)(14,220,122,111)(15,207,123,98)(16,222,124,85)(17,209,125,100)(18,224,126,87)(19,211,127,102)(20,198,128,89)(21,213,129,104)(22,200,130,91)(23,215,131,106)(24,202,132,93)(25,217,133,108)(26,204,134,95)(27,219,135,110)(28,206,136,97)(29,157,58,173)(30,144,59,188)(31,159,60,175)(32,146,61,190)(33,161,62,177)(34,148,63,192)(35,163,64,179)(36,150,65,194)(37,165,66,181)(38,152,67,196)(39,167,68,183)(40,154,69,170)(41,141,70,185)(42,156,71,172)(43,143,72,187)(44,158,73,174)(45,145,74,189)(46,160,75,176)(47,147,76,191)(48,162,77,178)(49,149,78,193)(50,164,79,180)(51,151,80,195)(52,166,81,182)(53,153,82,169)(54,168,83,184)(55,155,84,171)(56,142,57,186), (1,167,137,183)(2,168,138,184)(3,141,139,185)(4,142,140,186)(5,143,113,187)(6,144,114,188)(7,145,115,189)(8,146,116,190)(9,147,117,191)(10,148,118,192)(11,149,119,193)(12,150,120,194)(13,151,121,195)(14,152,122,196)(15,153,123,169)(16,154,124,170)(17,155,125,171)(18,156,126,172)(19,157,127,173)(20,158,128,174)(21,159,129,175)(22,160,130,176)(23,161,131,177)(24,162,132,178)(25,163,133,179)(26,164,134,180)(27,165,135,181)(28,166,136,182)(29,102,58,211)(30,103,59,212)(31,104,60,213)(32,105,61,214)(33,106,62,215)(34,107,63,216)(35,108,64,217)(36,109,65,218)(37,110,66,219)(38,111,67,220)(39,112,68,221)(40,85,69,222)(41,86,70,223)(42,87,71,224)(43,88,72,197)(44,89,73,198)(45,90,74,199)(46,91,75,200)(47,92,76,201)(48,93,77,202)(49,94,78,203)(50,95,79,204)(51,96,80,205)(52,97,81,206)(53,98,82,207)(54,99,83,208)(55,100,84,209)(56,101,57,210)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(104,112)(105,111)(106,110)(107,109)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(133,140)(134,139)(135,138)(136,137)(141,150)(142,149)(143,148)(144,147)(145,146)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(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), (1,221,137,112)(2,208,138,99)(3,223,139,86)(4,210,140,101)(5,197,113,88)(6,212,114,103)(7,199,115,90)(8,214,116,105)(9,201,117,92)(10,216,118,107)(11,203,119,94)(12,218,120,109)(13,205,121,96)(14,220,122,111)(15,207,123,98)(16,222,124,85)(17,209,125,100)(18,224,126,87)(19,211,127,102)(20,198,128,89)(21,213,129,104)(22,200,130,91)(23,215,131,106)(24,202,132,93)(25,217,133,108)(26,204,134,95)(27,219,135,110)(28,206,136,97)(29,157,58,173)(30,144,59,188)(31,159,60,175)(32,146,61,190)(33,161,62,177)(34,148,63,192)(35,163,64,179)(36,150,65,194)(37,165,66,181)(38,152,67,196)(39,167,68,183)(40,154,69,170)(41,141,70,185)(42,156,71,172)(43,143,72,187)(44,158,73,174)(45,145,74,189)(46,160,75,176)(47,147,76,191)(48,162,77,178)(49,149,78,193)(50,164,79,180)(51,151,80,195)(52,166,81,182)(53,153,82,169)(54,168,83,184)(55,155,84,171)(56,142,57,186), (1,167,137,183)(2,168,138,184)(3,141,139,185)(4,142,140,186)(5,143,113,187)(6,144,114,188)(7,145,115,189)(8,146,116,190)(9,147,117,191)(10,148,118,192)(11,149,119,193)(12,150,120,194)(13,151,121,195)(14,152,122,196)(15,153,123,169)(16,154,124,170)(17,155,125,171)(18,156,126,172)(19,157,127,173)(20,158,128,174)(21,159,129,175)(22,160,130,176)(23,161,131,177)(24,162,132,178)(25,163,133,179)(26,164,134,180)(27,165,135,181)(28,166,136,182)(29,102,58,211)(30,103,59,212)(31,104,60,213)(32,105,61,214)(33,106,62,215)(34,107,63,216)(35,108,64,217)(36,109,65,218)(37,110,66,219)(38,111,67,220)(39,112,68,221)(40,85,69,222)(41,86,70,223)(42,87,71,224)(43,88,72,197)(44,89,73,198)(45,90,74,199)(46,91,75,200)(47,92,76,201)(48,93,77,202)(49,94,78,203)(50,95,79,204)(51,96,80,205)(52,97,81,206)(53,98,82,207)(54,99,83,208)(55,100,84,209)(56,101,57,210) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(104,112),(105,111),(106,110),(107,109),(113,132),(114,131),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(121,124),(122,123),(133,140),(134,139),(135,138),(136,137),(141,150),(142,149),(143,148),(144,147),(145,146),(151,168),(152,167),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(169,182),(170,181),(171,180),(172,179),(173,178),(174,177),(175,176),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(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)], [(1,221,137,112),(2,208,138,99),(3,223,139,86),(4,210,140,101),(5,197,113,88),(6,212,114,103),(7,199,115,90),(8,214,116,105),(9,201,117,92),(10,216,118,107),(11,203,119,94),(12,218,120,109),(13,205,121,96),(14,220,122,111),(15,207,123,98),(16,222,124,85),(17,209,125,100),(18,224,126,87),(19,211,127,102),(20,198,128,89),(21,213,129,104),(22,200,130,91),(23,215,131,106),(24,202,132,93),(25,217,133,108),(26,204,134,95),(27,219,135,110),(28,206,136,97),(29,157,58,173),(30,144,59,188),(31,159,60,175),(32,146,61,190),(33,161,62,177),(34,148,63,192),(35,163,64,179),(36,150,65,194),(37,165,66,181),(38,152,67,196),(39,167,68,183),(40,154,69,170),(41,141,70,185),(42,156,71,172),(43,143,72,187),(44,158,73,174),(45,145,74,189),(46,160,75,176),(47,147,76,191),(48,162,77,178),(49,149,78,193),(50,164,79,180),(51,151,80,195),(52,166,81,182),(53,153,82,169),(54,168,83,184),(55,155,84,171),(56,142,57,186)], [(1,167,137,183),(2,168,138,184),(3,141,139,185),(4,142,140,186),(5,143,113,187),(6,144,114,188),(7,145,115,189),(8,146,116,190),(9,147,117,191),(10,148,118,192),(11,149,119,193),(12,150,120,194),(13,151,121,195),(14,152,122,196),(15,153,123,169),(16,154,124,170),(17,155,125,171),(18,156,126,172),(19,157,127,173),(20,158,128,174),(21,159,129,175),(22,160,130,176),(23,161,131,177),(24,162,132,178),(25,163,133,179),(26,164,134,180),(27,165,135,181),(28,166,136,182),(29,102,58,211),(30,103,59,212),(31,104,60,213),(32,105,61,214),(33,106,62,215),(34,107,63,216),(35,108,64,217),(36,109,65,218),(37,110,66,219),(38,111,67,220),(39,112,68,221),(40,85,69,222),(41,86,70,223),(42,87,71,224),(43,88,72,197),(44,89,73,198),(45,90,74,199),(46,91,75,200),(47,92,76,201),(48,93,77,202),(49,94,78,203),(50,95,79,204),(51,96,80,205),(52,97,81,206),(53,98,82,207),(54,99,83,208),(55,100,84,209),(56,101,57,210)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R | 28S | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D7 | SD16 | C4○D4 | D14 | D14 | C7⋊D4 | C8⋊C22 | Q8⋊D7 | Q8×D7 | Q8⋊2D7 | D4.D14 |
| kernel | D28⋊5Q8 | C28⋊C8 | C4.Dic14 | C14.D8 | C4×D28 | C7×C4⋊Q8 | D28 | C2×C28 | C4⋊Q8 | C28 | C28 | C42 | C4⋊C4 | C2×C4 | C14 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 3 | 4 | 2 | 3 | 6 | 12 | 1 | 6 | 3 | 3 | 6 |
Matrix representation of D28⋊5Q8 ►in GL6(𝔽113)
| 112 | 106 | 0 | 0 | 0 | 0 |
| 81 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 104 | 0 | 0 |
| 0 | 0 | 9 | 80 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 112 | 106 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 9 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 91 | 0 | 0 | 0 | 0 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 106 |
| 0 | 0 | 0 | 0 | 14 | 60 |
| 112 | 106 | 0 | 0 | 0 | 0 |
| 81 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 98 | 98 |
G:=sub<GL(6,GF(113))| [112,81,0,0,0,0,106,1,0,0,0,0,0,0,112,9,0,0,0,0,104,80,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,106,1,0,0,0,0,0,0,112,9,0,0,0,0,0,1,0,0,0,0,0,0,112,1,0,0,0,0,0,1],[0,36,0,0,0,0,91,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,14,0,0,0,0,106,60],[112,81,0,0,0,0,106,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,98,0,0,0,0,0,98] >;
D28⋊5Q8 in GAP, Magma, Sage, TeX
D_{28}\rtimes_5Q_8 % in TeX
G:=Group("D28:5Q8"); // GroupNames label
G:=SmallGroup(448,618);
// by ID
G=gap.SmallGroup(448,618);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,268,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations