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## G = C28⋊6SD16order 448 = 26·7

### 6th semidirect product of C28 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28⋊6SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C28⋊4D4 — C28⋊6SD16
 Lower central C7 — C14 — C2×C28 — C28⋊6SD16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C286SD16
G = < a,b,c | a28=b8=c2=1, bab-1=a13, cac=a-1, cbc=b3 >

Subgroups: 908 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C28, C28, D14, C2×C14, C4×C8, C41D4, C4⋊Q8, C2×SD16, C7⋊C8, D28, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C85D4, C2×C7⋊C8, Q8⋊D7, C4×C28, C7×C4⋊C4, C2×D28, C2×D28, Q8×C14, C4×C7⋊C8, C284D4, C2×Q8⋊D7, C7×C4⋊Q8, C286SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C41D4, C2×SD16, C7⋊D4, C22×D7, C85D4, Q8⋊D7, D4×D7, C2×C7⋊D4, C28⋊D4, C2×Q8⋊D7, C286SD16

Smallest permutation representation of C286SD16
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 81 209 118 149 102 177 33)(2 66 210 131 150 87 178 46)(3 79 211 116 151 100 179 31)(4 64 212 129 152 85 180 44)(5 77 213 114 153 98 181 29)(6 62 214 127 154 111 182 42)(7 75 215 140 155 96 183 55)(8 60 216 125 156 109 184 40)(9 73 217 138 157 94 185 53)(10 58 218 123 158 107 186 38)(11 71 219 136 159 92 187 51)(12 84 220 121 160 105 188 36)(13 69 221 134 161 90 189 49)(14 82 222 119 162 103 190 34)(15 67 223 132 163 88 191 47)(16 80 224 117 164 101 192 32)(17 65 197 130 165 86 193 45)(18 78 198 115 166 99 194 30)(19 63 199 128 167 112 195 43)(20 76 200 113 168 97 196 56)(21 61 201 126 141 110 169 41)(22 74 202 139 142 95 170 54)(23 59 203 124 143 108 171 39)(24 72 204 137 144 93 172 52)(25 57 205 122 145 106 173 37)(26 70 206 135 146 91 174 50)(27 83 207 120 147 104 175 35)(28 68 208 133 148 89 176 48)
(1 156)(2 155)(3 154)(4 153)(5 152)(6 151)(7 150)(8 149)(9 148)(10 147)(11 146)(12 145)(13 144)(14 143)(15 142)(16 141)(17 168)(18 167)(19 166)(20 165)(21 164)(22 163)(23 162)(24 161)(25 160)(26 159)(27 158)(28 157)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(85 114)(86 113)(87 140)(88 139)(89 138)(90 137)(91 136)(92 135)(93 134)(94 133)(95 132)(96 131)(97 130)(98 129)(99 128)(100 127)(101 126)(102 125)(103 124)(104 123)(105 122)(106 121)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(169 192)(170 191)(171 190)(172 189)(173 188)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(193 196)(194 195)(197 200)(198 199)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 7A 7B 7C 8A ··· 8H 14A ··· 14I 28A ··· 28R 28S ··· 28AD order 1 2 2 2 2 2 4 ··· 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 56 56 2 ··· 2 8 8 2 2 2 14 ··· 14 2 ··· 2 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 SD16 D14 D14 C7⋊D4 Q8⋊D7 D4×D7 kernel C28⋊6SD16 C4×C7⋊C8 C28⋊4D4 C2×Q8⋊D7 C7×C4⋊Q8 C7⋊C8 C2×C28 C4⋊Q8 C28 C42 C2×Q8 C2×C4 C4 C4 # reps 1 1 1 4 1 4 2 3 8 3 6 12 12 6

Matrix representation of C286SD16 in GL6(𝔽113)

 112 69 0 0 0 0 36 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 80 34 0 0 0 0 71 9
,
 0 106 0 0 0 0 16 26 0 0 0 0 0 0 100 100 0 0 0 0 13 100 0 0 0 0 0 0 60 22 0 0 0 0 88 53
,
 1 44 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 80 112 0 0 0 0 71 33

G:=sub<GL(6,GF(113))| [112,36,0,0,0,0,69,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,80,71,0,0,0,0,34,9],[0,16,0,0,0,0,106,26,0,0,0,0,0,0,100,13,0,0,0,0,100,100,0,0,0,0,0,0,60,88,0,0,0,0,22,53],[1,0,0,0,0,0,44,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,80,71,0,0,0,0,112,33] >;

C286SD16 in GAP, Magma, Sage, TeX

C_{28}\rtimes_6{\rm SD}_{16}
% in TeX

G:=Group("C28:6SD16");
// GroupNames label

G:=SmallGroup(448,619);
// by ID

G=gap.SmallGroup(448,619);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,219,184,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^13,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations

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