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## G = D28.30D4order 448 = 26·7

### 13rd non-split extension by D28 of D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D28.30D4
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C4○D28 — Q8.10D14 — D28.30D4
 Lower central C7 — C14 — C28 — D28.30D4
 Upper central C1 — C2 — C2×C4 — C2×Q16

Generators and relations for D28.30D4
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 >

Subgroups: 1156 in 248 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, Q16, Q16, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C8○D4, C2×Q16, C2×Q16, C4○D8, C8.C22, 2- 1+4, C7⋊C8, C56, Dic14, Dic14, Dic14, C4×D7, C4×D7, D28, D28, D28, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, Q8○D8, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, Q8⋊D7, C7⋊Q16, C2×C56, C7×Q16, C4○D28, C4○D28, C4○D28, Q8×D7, Q8×D7, Q82D7, Q82D7, Q8×C14, D28.2C4, D567C2, D7×Q16, Q16⋊D7, Q8.D14, C28.C23, C14×Q16, Q8.10D14, D28.30D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C22×D7, Q8○D8, D4×D7, C23×D7, C2×D4×D7, D28.30D4

Smallest permutation representation of D28.30D4
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 31)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(85 99)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(141 149)(142 148)(143 147)(144 146)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(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)(197 211)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 67 124 201 15 81 138 215)(2 68 125 202 16 82 139 216)(3 69 126 203 17 83 140 217)(4 70 127 204 18 84 113 218)(5 71 128 205 19 57 114 219)(6 72 129 206 20 58 115 220)(7 73 130 207 21 59 116 221)(8 74 131 208 22 60 117 222)(9 75 132 209 23 61 118 223)(10 76 133 210 24 62 119 224)(11 77 134 211 25 63 120 197)(12 78 135 212 26 64 121 198)(13 79 136 213 27 65 122 199)(14 80 137 214 28 66 123 200)(29 190 91 144 43 176 105 158)(30 191 92 145 44 177 106 159)(31 192 93 146 45 178 107 160)(32 193 94 147 46 179 108 161)(33 194 95 148 47 180 109 162)(34 195 96 149 48 181 110 163)(35 196 97 150 49 182 111 164)(36 169 98 151 50 183 112 165)(37 170 99 152 51 184 85 166)(38 171 100 153 52 185 86 167)(39 172 101 154 53 186 87 168)(40 173 102 155 54 187 88 141)(41 174 103 156 55 188 89 142)(42 175 104 157 56 189 90 143)
(1 149 15 163)(2 164 16 150)(3 151 17 165)(4 166 18 152)(5 153 19 167)(6 168 20 154)(7 155 21 141)(8 142 22 156)(9 157 23 143)(10 144 24 158)(11 159 25 145)(12 146 26 160)(13 161 27 147)(14 148 28 162)(29 224 43 210)(30 211 44 197)(31 198 45 212)(32 213 46 199)(33 200 47 214)(34 215 48 201)(35 202 49 216)(36 217 50 203)(37 204 51 218)(38 219 52 205)(39 206 53 220)(40 221 54 207)(41 208 55 222)(42 223 56 209)(57 86 71 100)(58 101 72 87)(59 88 73 102)(60 103 74 89)(61 90 75 104)(62 105 76 91)(63 92 77 106)(64 107 78 93)(65 94 79 108)(66 109 80 95)(67 96 81 110)(68 111 82 97)(69 98 83 112)(70 85 84 99)(113 170 127 184)(114 185 128 171)(115 172 129 186)(116 187 130 173)(117 174 131 188)(118 189 132 175)(119 176 133 190)(120 191 134 177)(121 178 135 192)(122 193 136 179)(123 180 137 194)(124 195 138 181)(125 182 139 196)(126 169 140 183)```

`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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,31)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(141,149)(142,148)(143,147)(144,146)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(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)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,67,124,201,15,81,138,215)(2,68,125,202,16,82,139,216)(3,69,126,203,17,83,140,217)(4,70,127,204,18,84,113,218)(5,71,128,205,19,57,114,219)(6,72,129,206,20,58,115,220)(7,73,130,207,21,59,116,221)(8,74,131,208,22,60,117,222)(9,75,132,209,23,61,118,223)(10,76,133,210,24,62,119,224)(11,77,134,211,25,63,120,197)(12,78,135,212,26,64,121,198)(13,79,136,213,27,65,122,199)(14,80,137,214,28,66,123,200)(29,190,91,144,43,176,105,158)(30,191,92,145,44,177,106,159)(31,192,93,146,45,178,107,160)(32,193,94,147,46,179,108,161)(33,194,95,148,47,180,109,162)(34,195,96,149,48,181,110,163)(35,196,97,150,49,182,111,164)(36,169,98,151,50,183,112,165)(37,170,99,152,51,184,85,166)(38,171,100,153,52,185,86,167)(39,172,101,154,53,186,87,168)(40,173,102,155,54,187,88,141)(41,174,103,156,55,188,89,142)(42,175,104,157,56,189,90,143), (1,149,15,163)(2,164,16,150)(3,151,17,165)(4,166,18,152)(5,153,19,167)(6,168,20,154)(7,155,21,141)(8,142,22,156)(9,157,23,143)(10,144,24,158)(11,159,25,145)(12,146,26,160)(13,161,27,147)(14,148,28,162)(29,224,43,210)(30,211,44,197)(31,198,45,212)(32,213,46,199)(33,200,47,214)(34,215,48,201)(35,202,49,216)(36,217,50,203)(37,204,51,218)(38,219,52,205)(39,206,53,220)(40,221,54,207)(41,208,55,222)(42,223,56,209)(57,86,71,100)(58,101,72,87)(59,88,73,102)(60,103,74,89)(61,90,75,104)(62,105,76,91)(63,92,77,106)(64,107,78,93)(65,94,79,108)(66,109,80,95)(67,96,81,110)(68,111,82,97)(69,98,83,112)(70,85,84,99)(113,170,127,184)(114,185,128,171)(115,172,129,186)(116,187,130,173)(117,174,131,188)(118,189,132,175)(119,176,133,190)(120,191,134,177)(121,178,135,192)(122,193,136,179)(123,180,137,194)(124,195,138,181)(125,182,139,196)(126,169,140,183)>;`

`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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,31)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(85,99)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(141,149)(142,148)(143,147)(144,146)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(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)(197,211)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,67,124,201,15,81,138,215)(2,68,125,202,16,82,139,216)(3,69,126,203,17,83,140,217)(4,70,127,204,18,84,113,218)(5,71,128,205,19,57,114,219)(6,72,129,206,20,58,115,220)(7,73,130,207,21,59,116,221)(8,74,131,208,22,60,117,222)(9,75,132,209,23,61,118,223)(10,76,133,210,24,62,119,224)(11,77,134,211,25,63,120,197)(12,78,135,212,26,64,121,198)(13,79,136,213,27,65,122,199)(14,80,137,214,28,66,123,200)(29,190,91,144,43,176,105,158)(30,191,92,145,44,177,106,159)(31,192,93,146,45,178,107,160)(32,193,94,147,46,179,108,161)(33,194,95,148,47,180,109,162)(34,195,96,149,48,181,110,163)(35,196,97,150,49,182,111,164)(36,169,98,151,50,183,112,165)(37,170,99,152,51,184,85,166)(38,171,100,153,52,185,86,167)(39,172,101,154,53,186,87,168)(40,173,102,155,54,187,88,141)(41,174,103,156,55,188,89,142)(42,175,104,157,56,189,90,143), (1,149,15,163)(2,164,16,150)(3,151,17,165)(4,166,18,152)(5,153,19,167)(6,168,20,154)(7,155,21,141)(8,142,22,156)(9,157,23,143)(10,144,24,158)(11,159,25,145)(12,146,26,160)(13,161,27,147)(14,148,28,162)(29,224,43,210)(30,211,44,197)(31,198,45,212)(32,213,46,199)(33,200,47,214)(34,215,48,201)(35,202,49,216)(36,217,50,203)(37,204,51,218)(38,219,52,205)(39,206,53,220)(40,221,54,207)(41,208,55,222)(42,223,56,209)(57,86,71,100)(58,101,72,87)(59,88,73,102)(60,103,74,89)(61,90,75,104)(62,105,76,91)(63,92,77,106)(64,107,78,93)(65,94,79,108)(66,109,80,95)(67,96,81,110)(68,111,82,97)(69,98,83,112)(70,85,84,99)(113,170,127,184)(114,185,128,171)(115,172,129,186)(116,187,130,173)(117,174,131,188)(118,189,132,175)(119,176,133,190)(120,191,134,177)(121,178,135,192)(122,193,136,179)(123,180,137,194)(124,195,138,181)(125,182,139,196)(126,169,140,183) );`

`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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,31),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,45),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(85,99),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(141,149),(142,148),(143,147),(144,146),(150,168),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(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),(197,211),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)], [(1,67,124,201,15,81,138,215),(2,68,125,202,16,82,139,216),(3,69,126,203,17,83,140,217),(4,70,127,204,18,84,113,218),(5,71,128,205,19,57,114,219),(6,72,129,206,20,58,115,220),(7,73,130,207,21,59,116,221),(8,74,131,208,22,60,117,222),(9,75,132,209,23,61,118,223),(10,76,133,210,24,62,119,224),(11,77,134,211,25,63,120,197),(12,78,135,212,26,64,121,198),(13,79,136,213,27,65,122,199),(14,80,137,214,28,66,123,200),(29,190,91,144,43,176,105,158),(30,191,92,145,44,177,106,159),(31,192,93,146,45,178,107,160),(32,193,94,147,46,179,108,161),(33,194,95,148,47,180,109,162),(34,195,96,149,48,181,110,163),(35,196,97,150,49,182,111,164),(36,169,98,151,50,183,112,165),(37,170,99,152,51,184,85,166),(38,171,100,153,52,185,86,167),(39,172,101,154,53,186,87,168),(40,173,102,155,54,187,88,141),(41,174,103,156,55,188,89,142),(42,175,104,157,56,189,90,143)], [(1,149,15,163),(2,164,16,150),(3,151,17,165),(4,166,18,152),(5,153,19,167),(6,168,20,154),(7,155,21,141),(8,142,22,156),(9,157,23,143),(10,144,24,158),(11,159,25,145),(12,146,26,160),(13,161,27,147),(14,148,28,162),(29,224,43,210),(30,211,44,197),(31,198,45,212),(32,213,46,199),(33,200,47,214),(34,215,48,201),(35,202,49,216),(36,217,50,203),(37,204,51,218),(38,219,52,205),(39,206,53,220),(40,221,54,207),(41,208,55,222),(42,223,56,209),(57,86,71,100),(58,101,72,87),(59,88,73,102),(60,103,74,89),(61,90,75,104),(62,105,76,91),(63,92,77,106),(64,107,78,93),(65,94,79,108),(66,109,80,95),(67,96,81,110),(68,111,82,97),(69,98,83,112),(70,85,84,99),(113,170,127,184),(114,185,128,171),(115,172,129,186),(116,187,130,173),(117,174,131,188),(118,189,132,175),(119,176,133,190),(120,191,134,177),(121,178,135,192),(122,193,136,179),(123,180,137,194),(124,195,138,181),(125,182,139,196),(126,169,140,183)]])`

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

Matrix representation of D28.30D4 in 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] >;`

D28.30D4 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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