metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊7Q8, C42.149D14, C14.1312+ (1+4), C28⋊Q8⋊36C2, C4.16(Q8×D7), C7⋊7(D4⋊3Q8), C28.51(C2×Q8), C42.C2⋊5D7, C4⋊C4.205D14, (C4×D28).24C2, D14.11(C2×Q8), D14⋊2Q8⋊35C2, D14⋊Q8⋊34C2, (C4×Dic14)⋊47C2, (C2×C28).88C23, C4.Dic14⋊34C2, D28⋊C4.11C2, C14.43(C22×Q8), (C2×C14).234C24, (C4×C28).194C22, D14⋊C4.40C22, C2.56(D4⋊8D14), Dic7.29(C4○D4), (C2×D28).267C22, C4⋊Dic7.379C22, C22.255(C23×D7), Dic7⋊C4.144C22, (C4×Dic7).141C22, (C2×Dic7).122C23, (C22×D7).221C23, (C2×Dic14).251C22, (D7×C4⋊C4)⋊35C2, C2.26(C2×Q8×D7), C2.85(D7×C4○D4), (C7×C42.C2)⋊7C2, C14.196(C2×C4○D4), (C2×C4×D7).217C22, (C2×C4).78(C22×D7), (C7×C4⋊C4).189C22, SmallGroup(448,1143)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1052 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×8], C7, C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×4], Q8 [×4], C23 [×2], D7 [×4], C14 [×3], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×6], C2×D4, C2×Q8 [×3], Dic7 [×2], Dic7 [×5], C28 [×2], C28 [×6], D14 [×4], D14 [×4], C2×C14, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2, C42.C2, C4⋊Q8, Dic14 [×4], C4×D7 [×8], D28 [×4], C2×Dic7 [×4], C2×Dic7 [×2], C2×C28 [×3], C2×C28 [×4], C22×D7 [×2], D4⋊3Q8, C4×Dic7 [×2], Dic7⋊C4 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4 [×6], C4×C28, C7×C4⋊C4 [×2], C7×C4⋊C4 [×4], C2×Dic14, C2×Dic14 [×2], C2×C4×D7 [×6], C2×D28, C4×Dic14, C4×D28, C28⋊Q8, C4.Dic14, D7×C4⋊C4 [×2], D28⋊C4 [×2], D14⋊Q8 [×4], D14⋊2Q8 [×2], C7×C42.C2, D28⋊7Q8
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊3Q8, Q8×D7 [×2], C23×D7, C2×Q8×D7, D7×C4○D4, D4⋊8D14, D28⋊7Q8
Generators and relations
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a13, cbc-1=a14b, dbd-1=a26b, 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 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 88)(13 87)(14 86)(15 85)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 159)(30 158)(31 157)(32 156)(33 155)(34 154)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 168)(49 167)(50 166)(51 165)(52 164)(53 163)(54 162)(55 161)(56 160)(57 214)(58 213)(59 212)(60 211)(61 210)(62 209)(63 208)(64 207)(65 206)(66 205)(67 204)(68 203)(69 202)(70 201)(71 200)(72 199)(73 198)(74 197)(75 224)(76 223)(77 222)(78 221)(79 220)(80 219)(81 218)(82 217)(83 216)(84 215)(113 196)(114 195)(115 194)(116 193)(117 192)(118 191)(119 190)(120 189)(121 188)(122 187)(123 186)(124 185)(125 184)(126 183)(127 182)(128 181)(129 180)(130 179)(131 178)(132 177)(133 176)(134 175)(135 174)(136 173)(137 172)(138 171)(139 170)(140 169)
(1 58 107 207)(2 59 108 208)(3 60 109 209)(4 61 110 210)(5 62 111 211)(6 63 112 212)(7 64 85 213)(8 65 86 214)(9 66 87 215)(10 67 88 216)(11 68 89 217)(12 69 90 218)(13 70 91 219)(14 71 92 220)(15 72 93 221)(16 73 94 222)(17 74 95 223)(18 75 96 224)(19 76 97 197)(20 77 98 198)(21 78 99 199)(22 79 100 200)(23 80 101 201)(24 81 102 202)(25 82 103 203)(26 83 104 204)(27 84 105 205)(28 57 106 206)(29 171 165 130)(30 172 166 131)(31 173 167 132)(32 174 168 133)(33 175 141 134)(34 176 142 135)(35 177 143 136)(36 178 144 137)(37 179 145 138)(38 180 146 139)(39 181 147 140)(40 182 148 113)(41 183 149 114)(42 184 150 115)(43 185 151 116)(44 186 152 117)(45 187 153 118)(46 188 154 119)(47 189 155 120)(48 190 156 121)(49 191 157 122)(50 192 158 123)(51 193 159 124)(52 194 160 125)(53 195 161 126)(54 196 162 127)(55 169 163 128)(56 170 164 129)
(1 117 107 186)(2 130 108 171)(3 115 109 184)(4 128 110 169)(5 113 111 182)(6 126 112 195)(7 139 85 180)(8 124 86 193)(9 137 87 178)(10 122 88 191)(11 135 89 176)(12 120 90 189)(13 133 91 174)(14 118 92 187)(15 131 93 172)(16 116 94 185)(17 129 95 170)(18 114 96 183)(19 127 97 196)(20 140 98 181)(21 125 99 194)(22 138 100 179)(23 123 101 192)(24 136 102 177)(25 121 103 190)(26 134 104 175)(27 119 105 188)(28 132 106 173)(29 59 165 208)(30 72 166 221)(31 57 167 206)(32 70 168 219)(33 83 141 204)(34 68 142 217)(35 81 143 202)(36 66 144 215)(37 79 145 200)(38 64 146 213)(39 77 147 198)(40 62 148 211)(41 75 149 224)(42 60 150 209)(43 73 151 222)(44 58 152 207)(45 71 153 220)(46 84 154 205)(47 69 155 218)(48 82 156 203)(49 67 157 216)(50 80 158 201)(51 65 159 214)(52 78 160 199)(53 63 161 212)(54 76 162 197)(55 61 163 210)(56 74 164 223)
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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,168)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,160)(57,214)(58,213)(59,212)(60,211)(61,210)(62,209)(63,208)(64,207)(65,206)(66,205)(67,204)(68,203)(69,202)(70,201)(71,200)(72,199)(73,198)(74,197)(75,224)(76,223)(77,222)(78,221)(79,220)(80,219)(81,218)(82,217)(83,216)(84,215)(113,196)(114,195)(115,194)(116,193)(117,192)(118,191)(119,190)(120,189)(121,188)(122,187)(123,186)(124,185)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169), (1,58,107,207)(2,59,108,208)(3,60,109,209)(4,61,110,210)(5,62,111,211)(6,63,112,212)(7,64,85,213)(8,65,86,214)(9,66,87,215)(10,67,88,216)(11,68,89,217)(12,69,90,218)(13,70,91,219)(14,71,92,220)(15,72,93,221)(16,73,94,222)(17,74,95,223)(18,75,96,224)(19,76,97,197)(20,77,98,198)(21,78,99,199)(22,79,100,200)(23,80,101,201)(24,81,102,202)(25,82,103,203)(26,83,104,204)(27,84,105,205)(28,57,106,206)(29,171,165,130)(30,172,166,131)(31,173,167,132)(32,174,168,133)(33,175,141,134)(34,176,142,135)(35,177,143,136)(36,178,144,137)(37,179,145,138)(38,180,146,139)(39,181,147,140)(40,182,148,113)(41,183,149,114)(42,184,150,115)(43,185,151,116)(44,186,152,117)(45,187,153,118)(46,188,154,119)(47,189,155,120)(48,190,156,121)(49,191,157,122)(50,192,158,123)(51,193,159,124)(52,194,160,125)(53,195,161,126)(54,196,162,127)(55,169,163,128)(56,170,164,129), (1,117,107,186)(2,130,108,171)(3,115,109,184)(4,128,110,169)(5,113,111,182)(6,126,112,195)(7,139,85,180)(8,124,86,193)(9,137,87,178)(10,122,88,191)(11,135,89,176)(12,120,90,189)(13,133,91,174)(14,118,92,187)(15,131,93,172)(16,116,94,185)(17,129,95,170)(18,114,96,183)(19,127,97,196)(20,140,98,181)(21,125,99,194)(22,138,100,179)(23,123,101,192)(24,136,102,177)(25,121,103,190)(26,134,104,175)(27,119,105,188)(28,132,106,173)(29,59,165,208)(30,72,166,221)(31,57,167,206)(32,70,168,219)(33,83,141,204)(34,68,142,217)(35,81,143,202)(36,66,144,215)(37,79,145,200)(38,64,146,213)(39,77,147,198)(40,62,148,211)(41,75,149,224)(42,60,150,209)(43,73,151,222)(44,58,152,207)(45,71,153,220)(46,84,154,205)(47,69,155,218)(48,82,156,203)(49,67,157,216)(50,80,158,201)(51,65,159,214)(52,78,160,199)(53,63,161,212)(54,76,162,197)(55,61,163,210)(56,74,164,223)>;
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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,88)(13,87)(14,86)(15,85)(16,112)(17,111)(18,110)(19,109)(20,108)(21,107)(22,106)(23,105)(24,104)(25,103)(26,102)(27,101)(28,100)(29,159)(30,158)(31,157)(32,156)(33,155)(34,154)(35,153)(36,152)(37,151)(38,150)(39,149)(40,148)(41,147)(42,146)(43,145)(44,144)(45,143)(46,142)(47,141)(48,168)(49,167)(50,166)(51,165)(52,164)(53,163)(54,162)(55,161)(56,160)(57,214)(58,213)(59,212)(60,211)(61,210)(62,209)(63,208)(64,207)(65,206)(66,205)(67,204)(68,203)(69,202)(70,201)(71,200)(72,199)(73,198)(74,197)(75,224)(76,223)(77,222)(78,221)(79,220)(80,219)(81,218)(82,217)(83,216)(84,215)(113,196)(114,195)(115,194)(116,193)(117,192)(118,191)(119,190)(120,189)(121,188)(122,187)(123,186)(124,185)(125,184)(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)(135,174)(136,173)(137,172)(138,171)(139,170)(140,169), (1,58,107,207)(2,59,108,208)(3,60,109,209)(4,61,110,210)(5,62,111,211)(6,63,112,212)(7,64,85,213)(8,65,86,214)(9,66,87,215)(10,67,88,216)(11,68,89,217)(12,69,90,218)(13,70,91,219)(14,71,92,220)(15,72,93,221)(16,73,94,222)(17,74,95,223)(18,75,96,224)(19,76,97,197)(20,77,98,198)(21,78,99,199)(22,79,100,200)(23,80,101,201)(24,81,102,202)(25,82,103,203)(26,83,104,204)(27,84,105,205)(28,57,106,206)(29,171,165,130)(30,172,166,131)(31,173,167,132)(32,174,168,133)(33,175,141,134)(34,176,142,135)(35,177,143,136)(36,178,144,137)(37,179,145,138)(38,180,146,139)(39,181,147,140)(40,182,148,113)(41,183,149,114)(42,184,150,115)(43,185,151,116)(44,186,152,117)(45,187,153,118)(46,188,154,119)(47,189,155,120)(48,190,156,121)(49,191,157,122)(50,192,158,123)(51,193,159,124)(52,194,160,125)(53,195,161,126)(54,196,162,127)(55,169,163,128)(56,170,164,129), (1,117,107,186)(2,130,108,171)(3,115,109,184)(4,128,110,169)(5,113,111,182)(6,126,112,195)(7,139,85,180)(8,124,86,193)(9,137,87,178)(10,122,88,191)(11,135,89,176)(12,120,90,189)(13,133,91,174)(14,118,92,187)(15,131,93,172)(16,116,94,185)(17,129,95,170)(18,114,96,183)(19,127,97,196)(20,140,98,181)(21,125,99,194)(22,138,100,179)(23,123,101,192)(24,136,102,177)(25,121,103,190)(26,134,104,175)(27,119,105,188)(28,132,106,173)(29,59,165,208)(30,72,166,221)(31,57,167,206)(32,70,168,219)(33,83,141,204)(34,68,142,217)(35,81,143,202)(36,66,144,215)(37,79,145,200)(38,64,146,213)(39,77,147,198)(40,62,148,211)(41,75,149,224)(42,60,150,209)(43,73,151,222)(44,58,152,207)(45,71,153,220)(46,84,154,205)(47,69,155,218)(48,82,156,203)(49,67,157,216)(50,80,158,201)(51,65,159,214)(52,78,160,199)(53,63,161,212)(54,76,162,197)(55,61,163,210)(56,74,164,223) );
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)], [(1,99),(2,98),(3,97),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,88),(13,87),(14,86),(15,85),(16,112),(17,111),(18,110),(19,109),(20,108),(21,107),(22,106),(23,105),(24,104),(25,103),(26,102),(27,101),(28,100),(29,159),(30,158),(31,157),(32,156),(33,155),(34,154),(35,153),(36,152),(37,151),(38,150),(39,149),(40,148),(41,147),(42,146),(43,145),(44,144),(45,143),(46,142),(47,141),(48,168),(49,167),(50,166),(51,165),(52,164),(53,163),(54,162),(55,161),(56,160),(57,214),(58,213),(59,212),(60,211),(61,210),(62,209),(63,208),(64,207),(65,206),(66,205),(67,204),(68,203),(69,202),(70,201),(71,200),(72,199),(73,198),(74,197),(75,224),(76,223),(77,222),(78,221),(79,220),(80,219),(81,218),(82,217),(83,216),(84,215),(113,196),(114,195),(115,194),(116,193),(117,192),(118,191),(119,190),(120,189),(121,188),(122,187),(123,186),(124,185),(125,184),(126,183),(127,182),(128,181),(129,180),(130,179),(131,178),(132,177),(133,176),(134,175),(135,174),(136,173),(137,172),(138,171),(139,170),(140,169)], [(1,58,107,207),(2,59,108,208),(3,60,109,209),(4,61,110,210),(5,62,111,211),(6,63,112,212),(7,64,85,213),(8,65,86,214),(9,66,87,215),(10,67,88,216),(11,68,89,217),(12,69,90,218),(13,70,91,219),(14,71,92,220),(15,72,93,221),(16,73,94,222),(17,74,95,223),(18,75,96,224),(19,76,97,197),(20,77,98,198),(21,78,99,199),(22,79,100,200),(23,80,101,201),(24,81,102,202),(25,82,103,203),(26,83,104,204),(27,84,105,205),(28,57,106,206),(29,171,165,130),(30,172,166,131),(31,173,167,132),(32,174,168,133),(33,175,141,134),(34,176,142,135),(35,177,143,136),(36,178,144,137),(37,179,145,138),(38,180,146,139),(39,181,147,140),(40,182,148,113),(41,183,149,114),(42,184,150,115),(43,185,151,116),(44,186,152,117),(45,187,153,118),(46,188,154,119),(47,189,155,120),(48,190,156,121),(49,191,157,122),(50,192,158,123),(51,193,159,124),(52,194,160,125),(53,195,161,126),(54,196,162,127),(55,169,163,128),(56,170,164,129)], [(1,117,107,186),(2,130,108,171),(3,115,109,184),(4,128,110,169),(5,113,111,182),(6,126,112,195),(7,139,85,180),(8,124,86,193),(9,137,87,178),(10,122,88,191),(11,135,89,176),(12,120,90,189),(13,133,91,174),(14,118,92,187),(15,131,93,172),(16,116,94,185),(17,129,95,170),(18,114,96,183),(19,127,97,196),(20,140,98,181),(21,125,99,194),(22,138,100,179),(23,123,101,192),(24,136,102,177),(25,121,103,190),(26,134,104,175),(27,119,105,188),(28,132,106,173),(29,59,165,208),(30,72,166,221),(31,57,167,206),(32,70,168,219),(33,83,141,204),(34,68,142,217),(35,81,143,202),(36,66,144,215),(37,79,145,200),(38,64,146,213),(39,77,147,198),(40,62,148,211),(41,75,149,224),(42,60,150,209),(43,73,151,222),(44,58,152,207),(45,71,153,220),(46,84,154,205),(47,69,155,218),(48,82,156,203),(49,67,157,216),(50,80,158,201),(51,65,159,214),(52,78,160,199),(53,63,161,212),(54,76,162,197),(55,61,163,210),(56,74,164,223)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 28 |
| 0 | 0 | 0 | 0 | 24 | 9 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 3 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 14 |
| 0 | 0 | 0 | 0 | 10 | 27 |
G:=sub<GL(6,GF(29))| [0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,3,0,0,0,0,26,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,26,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,24,0,0,0,0,28,9],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,0,0,2,10,0,0,0,0,14,27] >;
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 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | 2+ (1+4) | Q8×D7 | D7×C4○D4 | D4⋊8D14 |
| kernel | D28⋊7Q8 | C4×Dic14 | C4×D28 | C28⋊Q8 | C4.Dic14 | D7×C4⋊C4 | D28⋊C4 | D14⋊Q8 | D14⋊2Q8 | C7×C42.C2 | D28 | C42.C2 | Dic7 | C42 | C4⋊C4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 4 | 3 | 4 | 3 | 18 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_7Q_8 % in TeX
G:=Group("D28:7Q8"); // GroupNames label
G:=SmallGroup(448,1143);
// by ID
G=gap.SmallGroup(448,1143);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,184,1571,297,80,18822]);
// 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^13,c*b*c^-1=a^14*b,d*b*d^-1=a^26*b,d*c*d^-1=c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀