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## G = C28⋊5SD16order 448 = 26·7

### 5th semidirect product of C28 and SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28⋊5SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×D28 — C4×D28 — C28⋊5SD16
 Lower central C7 — C14 — C2×C28 — C28⋊5SD16
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C285SD16
G = < a,b,c | a28=b8=c2=1, bab-1=a-1, cac=a13, cbc=b3 >

Subgroups: 620 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, D14, C2×C14, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C7⋊C8, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, D4.D4, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, Q8⋊D7, C4×C28, C7×C4⋊C4, C2×C4×D7, C2×D28, Q8×C14, C28⋊C8, Q8⋊Dic7, C4×D28, C2×Q8⋊D7, C7×C4⋊Q8, C285SD16
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C8.C22, C7⋊D4, C22×D7, D4.D4, Q8⋊D7, D4×D7, D42D7, C2×C7⋊D4, C282D4, C2×Q8⋊D7, C28.C23, C285SD16

Smallest permutation representation of C285SD16
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 31 143 210 65 140 108 180)(2 30 144 209 66 139 109 179)(3 29 145 208 67 138 110 178)(4 56 146 207 68 137 111 177)(5 55 147 206 69 136 112 176)(6 54 148 205 70 135 85 175)(7 53 149 204 71 134 86 174)(8 52 150 203 72 133 87 173)(9 51 151 202 73 132 88 172)(10 50 152 201 74 131 89 171)(11 49 153 200 75 130 90 170)(12 48 154 199 76 129 91 169)(13 47 155 198 77 128 92 196)(14 46 156 197 78 127 93 195)(15 45 157 224 79 126 94 194)(16 44 158 223 80 125 95 193)(17 43 159 222 81 124 96 192)(18 42 160 221 82 123 97 191)(19 41 161 220 83 122 98 190)(20 40 162 219 84 121 99 189)(21 39 163 218 57 120 100 188)(22 38 164 217 58 119 101 187)(23 37 165 216 59 118 102 186)(24 36 166 215 60 117 103 185)(25 35 167 214 61 116 104 184)(26 34 168 213 62 115 105 183)(27 33 141 212 63 114 106 182)(28 32 142 211 64 113 107 181)
(1 15)(2 28)(3 13)(4 26)(5 11)(6 24)(7 9)(8 22)(10 20)(12 18)(14 16)(17 27)(19 25)(21 23)(29 198)(30 211)(31 224)(32 209)(33 222)(34 207)(35 220)(36 205)(37 218)(38 203)(39 216)(40 201)(41 214)(42 199)(43 212)(44 197)(45 210)(46 223)(47 208)(48 221)(49 206)(50 219)(51 204)(52 217)(53 202)(54 215)(55 200)(56 213)(57 59)(58 72)(60 70)(61 83)(62 68)(63 81)(64 66)(65 79)(67 77)(69 75)(71 73)(74 84)(76 82)(78 80)(85 166)(86 151)(87 164)(88 149)(89 162)(90 147)(91 160)(92 145)(93 158)(94 143)(95 156)(96 141)(97 154)(98 167)(99 152)(100 165)(101 150)(102 163)(103 148)(104 161)(105 146)(106 159)(107 144)(108 157)(109 142)(110 155)(111 168)(112 153)(113 179)(114 192)(115 177)(116 190)(117 175)(118 188)(119 173)(120 186)(121 171)(122 184)(123 169)(124 182)(125 195)(126 180)(127 193)(128 178)(129 191)(130 176)(131 189)(132 174)(133 187)(134 172)(135 185)(136 170)(137 183)(138 196)(139 181)(140 194)

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

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

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

61 conjugacy classes

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

61 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D7 SD16 C4○D4 D14 D14 C7⋊D4 C8.C22 Q8⋊D7 D4×D7 D4⋊2D7 C28.C23 kernel C28⋊5SD16 C28⋊C8 Q8⋊Dic7 C4×D28 C2×Q8⋊D7 C7×C4⋊Q8 D28 C2×C28 C4⋊Q8 C28 C28 C42 C2×Q8 C2×C4 C14 C4 C4 C4 C2 # reps 1 1 2 1 2 1 2 2 3 4 2 3 6 12 1 6 3 3 6

Matrix representation of C285SD16 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 89 0 0 0 0 24 103 0 0 0 0 0 0 52 28 0 0 0 0 85 61
,
 26 36 0 0 0 0 22 0 0 0 0 0 0 0 84 106 0 0 0 0 88 29 0 0 0 0 0 0 0 112 0 0 0 0 112 0
,
 1 0 0 0 0 0 106 112 0 0 0 0 0 0 1 0 0 0 0 0 24 112 0 0 0 0 0 0 112 0 0 0 0 0 0 112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,89,103,0,0,0,0,0,0,52,85,0,0,0,0,28,61],[26,22,0,0,0,0,36,0,0,0,0,0,0,0,84,88,0,0,0,0,106,29,0,0,0,0,0,0,0,112,0,0,0,0,112,0],[1,106,0,0,0,0,0,112,0,0,0,0,0,0,1,24,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112] >;

C285SD16 in GAP, Magma, Sage, TeX

C_{28}\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C28:5SD16");
// GroupNames label

G:=SmallGroup(448,617);
// by ID

G=gap.SmallGroup(448,617);
# by ID

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

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

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