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G = C28⋊4SD16order 448 = 26·7

4th semidirect product of C28 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28⋊4SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C28⋊2Q8 — C28⋊4SD16
 Lower central C7 — C14 — C2×C28 — C28⋊4SD16
 Upper central C1 — C22 — C42 — C4⋊1D4

Generators and relations for C284SD16
G = < a,b,c | a28=b8=c2=1, bab-1=a13, cac=a15, cbc=b3 >

Subgroups: 620 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, C14, C14, C14, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C4×C8, C41D4, C4⋊Q8, C2×SD16, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C85D4, C2×C7⋊C8, C4⋊Dic7, D4.D7, C4×C28, C2×Dic14, D4×C14, D4×C14, C4×C7⋊C8, C282Q8, C2×D4.D7, C7×C41D4, C284SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C41D4, C2×SD16, C7⋊D4, C22×D7, C85D4, D4.D7, D4×D7, C2×C7⋊D4, C2×D4.D7, C28⋊D4, C284SD16

Smallest permutation representation of C284SD16
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 30 63 103 202 135 149 196)(2 43 64 88 203 120 150 181)(3 56 65 101 204 133 151 194)(4 41 66 86 205 118 152 179)(5 54 67 99 206 131 153 192)(6 39 68 112 207 116 154 177)(7 52 69 97 208 129 155 190)(8 37 70 110 209 114 156 175)(9 50 71 95 210 127 157 188)(10 35 72 108 211 140 158 173)(11 48 73 93 212 125 159 186)(12 33 74 106 213 138 160 171)(13 46 75 91 214 123 161 184)(14 31 76 104 215 136 162 169)(15 44 77 89 216 121 163 182)(16 29 78 102 217 134 164 195)(17 42 79 87 218 119 165 180)(18 55 80 100 219 132 166 193)(19 40 81 85 220 117 167 178)(20 53 82 98 221 130 168 191)(21 38 83 111 222 115 141 176)(22 51 84 96 223 128 142 189)(23 36 57 109 224 113 143 174)(24 49 58 94 197 126 144 187)(25 34 59 107 198 139 145 172)(26 47 60 92 199 124 146 185)(27 32 61 105 200 137 147 170)(28 45 62 90 201 122 148 183)
(1 223)(2 210)(3 197)(4 212)(5 199)(6 214)(7 201)(8 216)(9 203)(10 218)(11 205)(12 220)(13 207)(14 222)(15 209)(16 224)(17 211)(18 198)(19 213)(20 200)(21 215)(22 202)(23 217)(24 204)(25 219)(26 206)(27 221)(28 208)(29 174)(30 189)(31 176)(32 191)(33 178)(34 193)(35 180)(36 195)(37 182)(38 169)(39 184)(40 171)(41 186)(42 173)(43 188)(44 175)(45 190)(46 177)(47 192)(48 179)(49 194)(50 181)(51 196)(52 183)(53 170)(54 185)(55 172)(56 187)(57 78)(58 65)(59 80)(60 67)(61 82)(62 69)(63 84)(64 71)(66 73)(68 75)(70 77)(72 79)(74 81)(76 83)(85 138)(86 125)(87 140)(88 127)(89 114)(90 129)(91 116)(92 131)(93 118)(94 133)(95 120)(96 135)(97 122)(98 137)(99 124)(100 139)(101 126)(102 113)(103 128)(104 115)(105 130)(106 117)(107 132)(108 119)(109 134)(110 121)(111 136)(112 123)(141 162)(142 149)(143 164)(144 151)(145 166)(146 153)(147 168)(148 155)(150 157)(152 159)(154 161)(156 163)(158 165)(160 167)

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

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

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

64 conjugacy classes

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

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 D4.D7 D4×D7 kernel C28⋊4SD16 C4×C7⋊C8 C28⋊2Q8 C2×D4.D7 C7×C4⋊1D4 C7⋊C8 C2×C28 C4⋊1D4 C28 C42 C2×D4 C2×C4 C4 C4 # reps 1 1 1 4 1 4 2 3 8 3 6 12 12 6

Matrix representation of C284SD16 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 72 63 0 0 0 0 2 41 0 0 0 0 0 0 7 0 0 0 0 0 27 97
,
 87 7 0 0 0 0 97 0 0 0 0 0 0 0 112 0 0 0 0 0 0 112 0 0 0 0 0 0 77 106 0 0 0 0 40 36
,
 1 69 0 0 0 0 0 112 0 0 0 0 0 0 41 50 0 0 0 0 102 72 0 0 0 0 0 0 112 0 0 0 0 0 91 1

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,72,2,0,0,0,0,63,41,0,0,0,0,0,0,7,27,0,0,0,0,0,97],[87,97,0,0,0,0,7,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,77,40,0,0,0,0,106,36],[1,0,0,0,0,0,69,112,0,0,0,0,0,0,41,102,0,0,0,0,50,72,0,0,0,0,0,0,112,91,0,0,0,0,0,1] >;

C284SD16 in GAP, Magma, Sage, TeX

C_{28}\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C28:4SD16");
// GroupNames label

G:=SmallGroup(448,610);
// by ID

G=gap.SmallGroup(448,610);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,120,254,219,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^13,c*a*c=a^15,c*b*c=b^3>;
// generators/relations

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