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## G = C8.20D28order 448 = 26·7

### 6th non-split extension by C8 of D28 acting via D28/D14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C8.20D28
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D28.2C4 — C8.20D28
 Lower central C7 — C14 — C2×C28 — C8.20D28
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C8.20D28
G = < a,b,c | a56=1, b4=c2=a28, bab-1=a15, cac-1=a-1, cbc-1=b3 >

Subgroups: 540 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C7⋊C8, C56, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, D4.5D4, C8×D7, C8⋊D7, C56⋊C2, Dic28, C4.Dic7, C2×C56, C7×M4(2), C2×Dic14, C4○D28, C4.12D28, C7×C8.C4, D28.2C4, C2×Dic28, C8.D14, C8.20D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, D28, C22×D7, D4.5D4, C2×D28, D4×D7, Q82D7, C4⋊D28, C8.20D28

Smallest permutation representation of C8.20D28
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 84 15 70 29 112 43 98)(2 99 16 85 30 71 44 57)(3 58 17 100 31 86 45 72)(4 73 18 59 32 101 46 87)(5 88 19 74 33 60 47 102)(6 103 20 89 34 75 48 61)(7 62 21 104 35 90 49 76)(8 77 22 63 36 105 50 91)(9 92 23 78 37 64 51 106)(10 107 24 93 38 79 52 65)(11 66 25 108 39 94 53 80)(12 81 26 67 40 109 54 95)(13 96 27 82 41 68 55 110)(14 111 28 97 42 83 56 69)(113 209 127 195 141 181 155 223)(114 224 128 210 142 196 156 182)(115 183 129 169 143 211 157 197)(116 198 130 184 144 170 158 212)(117 213 131 199 145 185 159 171)(118 172 132 214 146 200 160 186)(119 187 133 173 147 215 161 201)(120 202 134 188 148 174 162 216)(121 217 135 203 149 189 163 175)(122 176 136 218 150 204 164 190)(123 191 137 177 151 219 165 205)(124 206 138 192 152 178 166 220)(125 221 139 207 153 193 167 179)(126 180 140 222 154 208 168 194)
(1 122 29 150)(2 121 30 149)(3 120 31 148)(4 119 32 147)(5 118 33 146)(6 117 34 145)(7 116 35 144)(8 115 36 143)(9 114 37 142)(10 113 38 141)(11 168 39 140)(12 167 40 139)(13 166 41 138)(14 165 42 137)(15 164 43 136)(16 163 44 135)(17 162 45 134)(18 161 46 133)(19 160 47 132)(20 159 48 131)(21 158 49 130)(22 157 50 129)(23 156 51 128)(24 155 52 127)(25 154 53 126)(26 153 54 125)(27 152 55 124)(28 151 56 123)(57 189 85 217)(58 188 86 216)(59 187 87 215)(60 186 88 214)(61 185 89 213)(62 184 90 212)(63 183 91 211)(64 182 92 210)(65 181 93 209)(66 180 94 208)(67 179 95 207)(68 178 96 206)(69 177 97 205)(70 176 98 204)(71 175 99 203)(72 174 100 202)(73 173 101 201)(74 172 102 200)(75 171 103 199)(76 170 104 198)(77 169 105 197)(78 224 106 196)(79 223 107 195)(80 222 108 194)(81 221 109 193)(82 220 110 192)(83 219 111 191)(84 218 112 190)

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L 56M ··· 56X order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 56 ··· 56 size 1 1 2 28 2 2 28 56 56 2 2 2 2 2 4 8 8 28 28 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4 8 ··· 8

58 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 C4○D4 D14 D14 D28 D4.5D4 D4×D7 Q8⋊2D7 C8.20D28 kernel C8.20D28 C4.12D28 C7×C8.C4 D28.2C4 C2×Dic28 C8.D14 C56 Dic14 D28 C8.C4 C2×C14 C2×C8 M4(2) C8 C7 C4 C22 C1 # reps 1 2 1 1 1 2 2 1 1 3 2 3 6 12 2 3 3 12

Matrix representation of C8.20D28 in GL4(𝔽113) generated by

 105 21 0 0 96 101 0 0 94 65 81 41 93 84 31 45
,
 102 38 105 20 71 104 65 22 88 35 66 75 52 84 68 67
,
 46 40 0 0 29 67 0 0 62 51 43 89 42 64 30 70
G:=sub<GL(4,GF(113))| [105,96,94,93,21,101,65,84,0,0,81,31,0,0,41,45],[102,71,88,52,38,104,35,84,105,65,66,68,20,22,75,67],[46,29,62,42,40,67,51,64,0,0,43,30,0,0,89,70] >;

C8.20D28 in GAP, Magma, Sage, TeX

C_8._{20}D_{28}
% in TeX

G:=Group("C8.20D28");
// GroupNames label

G:=SmallGroup(448,430);
// by ID

G=gap.SmallGroup(448,430);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,58,1123,136,438,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=1,b^4=c^2=a^28,b*a*b^-1=a^15,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

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