metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊9Q8, C42.175D14, C14.832+ (1+4), C4⋊Q8⋊13D7, C4.19(Q8×D7), C7⋊8(D4⋊3Q8), C28.56(C2×Q8), C4⋊C4.220D14, (C4×D28).27C2, (C2×Q8).87D14, D14.12(C2×Q8), D14⋊3Q8⋊37C2, D14⋊2Q8⋊44C2, (C4×Dic14)⋊53C2, C4.Dic14⋊44C2, D28⋊C4.14C2, C28.137(C4○D4), C14.50(C22×Q8), (C4×C28).215C22, (C2×C14).274C24, (C2×C28).107C23, C4.40(Q8⋊2D7), C2.87(D4⋊6D14), D14⋊C4.153C22, (C2×D28).273C22, Dic7⋊C4.62C22, C4⋊Dic7.253C22, (Q8×C14).141C22, C22.295(C23×D7), (C2×Dic7).145C23, (C4×Dic7).163C22, (C22×D7).235C23, (C2×Dic14).303C22, (D7×C4⋊C4)⋊45C2, C2.33(C2×Q8×D7), (C7×C4⋊Q8)⋊16C2, C14.122(C2×C4○D4), C2.30(C2×Q8⋊2D7), (C2×C4×D7).147C22, (C7×C4⋊C4).217C22, (C2×C4).220(C22×D7), SmallGroup(448,1183)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1004 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×11], 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 [×4], C4⋊C4 [×12], C22×C4 [×6], C2×D4, C2×Q8 [×2], C2×Q8, Dic7 [×6], C28 [×4], C28 [×5], D14 [×4], D14 [×4], C2×C14, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C42.C2 [×2], C4⋊Q8, Dic14 [×2], C4×D7 [×8], D28 [×4], C2×Dic7 [×6], C2×C28 [×3], C2×C28 [×4], C7×Q8 [×2], C22×D7 [×2], D4⋊3Q8, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], C4⋊Dic7 [×4], D14⋊C4 [×6], C4×C28, C7×C4⋊C4 [×4], C2×Dic14, C2×C4×D7 [×6], C2×D28, Q8×C14 [×2], C4×Dic14, C4×D28, C4.Dic14 [×2], D7×C4⋊C4 [×2], D28⋊C4 [×2], D14⋊2Q8 [×2], D14⋊3Q8 [×4], C7×C4⋊Q8, D28⋊9Q8
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], Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8×D7, C2×Q8⋊2D7, D28⋊9Q8
Generators and relations
G = < a,b,c,d | a28=b2=c4=1, d2=c2, bab=a-1, cac-1=a15, ad=da, cbc-1=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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 112)(18 111)(19 110)(20 109)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 149)(30 148)(31 147)(32 146)(33 145)(34 144)(35 143)(36 142)(37 141)(38 168)(39 167)(40 166)(41 165)(42 164)(43 163)(44 162)(45 161)(46 160)(47 159)(48 158)(49 157)(50 156)(51 155)(52 154)(53 153)(54 152)(55 151)(56 150)(57 134)(58 133)(59 132)(60 131)(61 130)(62 129)(63 128)(64 127)(65 126)(66 125)(67 124)(68 123)(69 122)(70 121)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 140)(80 139)(81 138)(82 137)(83 136)(84 135)(169 197)(170 224)(171 223)(172 222)(173 221)(174 220)(175 219)(176 218)(177 217)(178 216)(179 215)(180 214)(181 213)(182 212)(183 211)(184 210)(185 209)(186 208)(187 207)(188 206)(189 205)(190 204)(191 203)(192 202)(193 201)(194 200)(195 199)(196 198)
(1 113 108 72)(2 128 109 59)(3 115 110 74)(4 130 111 61)(5 117 112 76)(6 132 85 63)(7 119 86 78)(8 134 87 65)(9 121 88 80)(10 136 89 67)(11 123 90 82)(12 138 91 69)(13 125 92 84)(14 140 93 71)(15 127 94 58)(16 114 95 73)(17 129 96 60)(18 116 97 75)(19 131 98 62)(20 118 99 77)(21 133 100 64)(22 120 101 79)(23 135 102 66)(24 122 103 81)(25 137 104 68)(26 124 105 83)(27 139 106 70)(28 126 107 57)(29 170 141 202)(30 185 142 217)(31 172 143 204)(32 187 144 219)(33 174 145 206)(34 189 146 221)(35 176 147 208)(36 191 148 223)(37 178 149 210)(38 193 150 197)(39 180 151 212)(40 195 152 199)(41 182 153 214)(42 169 154 201)(43 184 155 216)(44 171 156 203)(45 186 157 218)(46 173 158 205)(47 188 159 220)(48 175 160 207)(49 190 161 222)(50 177 162 209)(51 192 163 224)(52 179 164 211)(53 194 165 198)(54 181 166 213)(55 196 167 200)(56 183 168 215)
(1 156 108 44)(2 157 109 45)(3 158 110 46)(4 159 111 47)(5 160 112 48)(6 161 85 49)(7 162 86 50)(8 163 87 51)(9 164 88 52)(10 165 89 53)(11 166 90 54)(12 167 91 55)(13 168 92 56)(14 141 93 29)(15 142 94 30)(16 143 95 31)(17 144 96 32)(18 145 97 33)(19 146 98 34)(20 147 99 35)(21 148 100 36)(22 149 101 37)(23 150 102 38)(24 151 103 39)(25 152 104 40)(26 153 105 41)(27 154 106 42)(28 155 107 43)(57 216 126 184)(58 217 127 185)(59 218 128 186)(60 219 129 187)(61 220 130 188)(62 221 131 189)(63 222 132 190)(64 223 133 191)(65 224 134 192)(66 197 135 193)(67 198 136 194)(68 199 137 195)(69 200 138 196)(70 201 139 169)(71 202 140 170)(72 203 113 171)(73 204 114 172)(74 205 115 173)(75 206 116 174)(76 207 117 175)(77 208 118 176)(78 209 119 177)(79 210 120 178)(80 211 121 179)(81 212 122 180)(82 213 123 181)(83 214 124 182)(84 215 125 183)
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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,168)(39,167)(40,166)(41,165)(42,164)(43,163)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(169,197)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198), (1,113,108,72)(2,128,109,59)(3,115,110,74)(4,130,111,61)(5,117,112,76)(6,132,85,63)(7,119,86,78)(8,134,87,65)(9,121,88,80)(10,136,89,67)(11,123,90,82)(12,138,91,69)(13,125,92,84)(14,140,93,71)(15,127,94,58)(16,114,95,73)(17,129,96,60)(18,116,97,75)(19,131,98,62)(20,118,99,77)(21,133,100,64)(22,120,101,79)(23,135,102,66)(24,122,103,81)(25,137,104,68)(26,124,105,83)(27,139,106,70)(28,126,107,57)(29,170,141,202)(30,185,142,217)(31,172,143,204)(32,187,144,219)(33,174,145,206)(34,189,146,221)(35,176,147,208)(36,191,148,223)(37,178,149,210)(38,193,150,197)(39,180,151,212)(40,195,152,199)(41,182,153,214)(42,169,154,201)(43,184,155,216)(44,171,156,203)(45,186,157,218)(46,173,158,205)(47,188,159,220)(48,175,160,207)(49,190,161,222)(50,177,162,209)(51,192,163,224)(52,179,164,211)(53,194,165,198)(54,181,166,213)(55,196,167,200)(56,183,168,215), (1,156,108,44)(2,157,109,45)(3,158,110,46)(4,159,111,47)(5,160,112,48)(6,161,85,49)(7,162,86,50)(8,163,87,51)(9,164,88,52)(10,165,89,53)(11,166,90,54)(12,167,91,55)(13,168,92,56)(14,141,93,29)(15,142,94,30)(16,143,95,31)(17,144,96,32)(18,145,97,33)(19,146,98,34)(20,147,99,35)(21,148,100,36)(22,149,101,37)(23,150,102,38)(24,151,103,39)(25,152,104,40)(26,153,105,41)(27,154,106,42)(28,155,107,43)(57,216,126,184)(58,217,127,185)(59,218,128,186)(60,219,129,187)(61,220,130,188)(62,221,131,189)(63,222,132,190)(64,223,133,191)(65,224,134,192)(66,197,135,193)(67,198,136,194)(68,199,137,195)(69,200,138,196)(70,201,139,169)(71,202,140,170)(72,203,113,171)(73,204,114,172)(74,205,115,173)(75,206,116,174)(76,207,117,175)(77,208,118,176)(78,209,119,177)(79,210,120,178)(80,211,121,179)(81,212,122,180)(82,213,123,181)(83,214,124,182)(84,215,125,183)>;
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), (1,100)(2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,149)(30,148)(31,147)(32,146)(33,145)(34,144)(35,143)(36,142)(37,141)(38,168)(39,167)(40,166)(41,165)(42,164)(43,163)(44,162)(45,161)(46,160)(47,159)(48,158)(49,157)(50,156)(51,155)(52,154)(53,153)(54,152)(55,151)(56,150)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,128)(64,127)(65,126)(66,125)(67,124)(68,123)(69,122)(70,121)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,140)(80,139)(81,138)(82,137)(83,136)(84,135)(169,197)(170,224)(171,223)(172,222)(173,221)(174,220)(175,219)(176,218)(177,217)(178,216)(179,215)(180,214)(181,213)(182,212)(183,211)(184,210)(185,209)(186,208)(187,207)(188,206)(189,205)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198), (1,113,108,72)(2,128,109,59)(3,115,110,74)(4,130,111,61)(5,117,112,76)(6,132,85,63)(7,119,86,78)(8,134,87,65)(9,121,88,80)(10,136,89,67)(11,123,90,82)(12,138,91,69)(13,125,92,84)(14,140,93,71)(15,127,94,58)(16,114,95,73)(17,129,96,60)(18,116,97,75)(19,131,98,62)(20,118,99,77)(21,133,100,64)(22,120,101,79)(23,135,102,66)(24,122,103,81)(25,137,104,68)(26,124,105,83)(27,139,106,70)(28,126,107,57)(29,170,141,202)(30,185,142,217)(31,172,143,204)(32,187,144,219)(33,174,145,206)(34,189,146,221)(35,176,147,208)(36,191,148,223)(37,178,149,210)(38,193,150,197)(39,180,151,212)(40,195,152,199)(41,182,153,214)(42,169,154,201)(43,184,155,216)(44,171,156,203)(45,186,157,218)(46,173,158,205)(47,188,159,220)(48,175,160,207)(49,190,161,222)(50,177,162,209)(51,192,163,224)(52,179,164,211)(53,194,165,198)(54,181,166,213)(55,196,167,200)(56,183,168,215), (1,156,108,44)(2,157,109,45)(3,158,110,46)(4,159,111,47)(5,160,112,48)(6,161,85,49)(7,162,86,50)(8,163,87,51)(9,164,88,52)(10,165,89,53)(11,166,90,54)(12,167,91,55)(13,168,92,56)(14,141,93,29)(15,142,94,30)(16,143,95,31)(17,144,96,32)(18,145,97,33)(19,146,98,34)(20,147,99,35)(21,148,100,36)(22,149,101,37)(23,150,102,38)(24,151,103,39)(25,152,104,40)(26,153,105,41)(27,154,106,42)(28,155,107,43)(57,216,126,184)(58,217,127,185)(59,218,128,186)(60,219,129,187)(61,220,130,188)(62,221,131,189)(63,222,132,190)(64,223,133,191)(65,224,134,192)(66,197,135,193)(67,198,136,194)(68,199,137,195)(69,200,138,196)(70,201,139,169)(71,202,140,170)(72,203,113,171)(73,204,114,172)(74,205,115,173)(75,206,116,174)(76,207,117,175)(77,208,118,176)(78,209,119,177)(79,210,120,178)(80,211,121,179)(81,212,122,180)(82,213,123,181)(83,214,124,182)(84,215,125,183) );
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)], [(1,100),(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,112),(18,111),(19,110),(20,109),(21,108),(22,107),(23,106),(24,105),(25,104),(26,103),(27,102),(28,101),(29,149),(30,148),(31,147),(32,146),(33,145),(34,144),(35,143),(36,142),(37,141),(38,168),(39,167),(40,166),(41,165),(42,164),(43,163),(44,162),(45,161),(46,160),(47,159),(48,158),(49,157),(50,156),(51,155),(52,154),(53,153),(54,152),(55,151),(56,150),(57,134),(58,133),(59,132),(60,131),(61,130),(62,129),(63,128),(64,127),(65,126),(66,125),(67,124),(68,123),(69,122),(70,121),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,140),(80,139),(81,138),(82,137),(83,136),(84,135),(169,197),(170,224),(171,223),(172,222),(173,221),(174,220),(175,219),(176,218),(177,217),(178,216),(179,215),(180,214),(181,213),(182,212),(183,211),(184,210),(185,209),(186,208),(187,207),(188,206),(189,205),(190,204),(191,203),(192,202),(193,201),(194,200),(195,199),(196,198)], [(1,113,108,72),(2,128,109,59),(3,115,110,74),(4,130,111,61),(5,117,112,76),(6,132,85,63),(7,119,86,78),(8,134,87,65),(9,121,88,80),(10,136,89,67),(11,123,90,82),(12,138,91,69),(13,125,92,84),(14,140,93,71),(15,127,94,58),(16,114,95,73),(17,129,96,60),(18,116,97,75),(19,131,98,62),(20,118,99,77),(21,133,100,64),(22,120,101,79),(23,135,102,66),(24,122,103,81),(25,137,104,68),(26,124,105,83),(27,139,106,70),(28,126,107,57),(29,170,141,202),(30,185,142,217),(31,172,143,204),(32,187,144,219),(33,174,145,206),(34,189,146,221),(35,176,147,208),(36,191,148,223),(37,178,149,210),(38,193,150,197),(39,180,151,212),(40,195,152,199),(41,182,153,214),(42,169,154,201),(43,184,155,216),(44,171,156,203),(45,186,157,218),(46,173,158,205),(47,188,159,220),(48,175,160,207),(49,190,161,222),(50,177,162,209),(51,192,163,224),(52,179,164,211),(53,194,165,198),(54,181,166,213),(55,196,167,200),(56,183,168,215)], [(1,156,108,44),(2,157,109,45),(3,158,110,46),(4,159,111,47),(5,160,112,48),(6,161,85,49),(7,162,86,50),(8,163,87,51),(9,164,88,52),(10,165,89,53),(11,166,90,54),(12,167,91,55),(13,168,92,56),(14,141,93,29),(15,142,94,30),(16,143,95,31),(17,144,96,32),(18,145,97,33),(19,146,98,34),(20,147,99,35),(21,148,100,36),(22,149,101,37),(23,150,102,38),(24,151,103,39),(25,152,104,40),(26,153,105,41),(27,154,106,42),(28,155,107,43),(57,216,126,184),(58,217,127,185),(59,218,128,186),(60,219,129,187),(61,220,130,188),(62,221,131,189),(63,222,132,190),(64,223,133,191),(65,224,134,192),(66,197,135,193),(67,198,136,194),(68,199,137,195),(69,200,138,196),(70,201,139,169),(71,202,140,170),(72,203,113,171),(73,204,114,172),(74,205,115,173),(75,206,116,174),(76,207,117,175),(77,208,118,176),(78,209,119,177),(79,210,120,178),(80,211,121,179),(81,212,122,180),(82,213,123,181),(83,214,124,182),(84,215,125,183)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 25 | 18 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 25 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 13 | 0 | 0 |
| 0 | 0 | 11 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 27 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 0 | 0 | 0 | 0 | 7 | 9 |
G:=sub<GL(6,GF(29))| [25,11,0,0,0,0,18,1,0,0,0,0,0,0,12,0,0,0,0,0,11,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,25,0,0,0,0,11,25,0,0,0,0,0,0,28,18,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,13,28,0,0,0,0,0,0,1,27,0,0,0,0,1,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,11,17,0,0,0,0,0,0,20,7,0,0,0,0,9,9] >;
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 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | 2+ (1+4) | Q8×D7 | Q8⋊2D7 | D4⋊6D14 |
| kernel | D28⋊9Q8 | C4×Dic14 | C4×D28 | C4.Dic14 | D7×C4⋊C4 | D28⋊C4 | D14⋊2Q8 | D14⋊3Q8 | C7×C4⋊Q8 | D28 | C4⋊Q8 | C28 | C42 | C4⋊C4 | C2×Q8 | C14 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 3 | 4 | 3 | 12 | 6 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
D_{28}\rtimes_9Q_8 % in TeX
G:=Group("D28:9Q8"); // GroupNames label
G:=SmallGroup(448,1183);
// by ID
G=gap.SmallGroup(448,1183);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,232,100,570,185,192,18822]);
// 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=d*b*d^-1=a^14*b,d*c*d^-1=c^-1>;
// generators/relations
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