metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28⋊1C8, C4.17D56, C28.35D8, C28.29SD16, C42.249D14, C28.13M4(2), (C4×C8)⋊1D7, (C4×C56)⋊1C2, C7⋊1(D4⋊C8), C4.7(C8×D7), C14.4C4≀C2, C28⋊C8⋊1C2, C28.17(C2×C8), (C4×D28).1C2, (C2×D28).5C4, C4⋊Dic7.5C4, C2.4(D14⋊C8), (C2×C4).161D28, (C2×C28).436D4, C4.5(C8⋊D7), C4.16(C56⋊C2), C14.2(C22⋊C8), C2.1(C2.D56), C2.2(Dic14⋊C4), (C4×C28).320C22, C14.10(D4⋊C4), C22.31(D14⋊C4), (C2×C4).96(C4×D7), (C2×C28).216(C2×C4), (C2×C4).205(C7⋊D4), (C2×C14).36(C22⋊C4), SmallGroup(448,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.17D56
G = < a,b,c | a4=b56=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
Subgroups: 484 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C4×C8, C4⋊C8, C4×D4, C7⋊C8, C56, C4×D7, D28, D28, C2×Dic7, C2×C28, C22×D7, D4⋊C8, C2×C7⋊C8, C4⋊Dic7, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C2×D28, C28⋊C8, C4×C56, C4×D28, C4.17D56
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), D8, SD16, D14, C22⋊C8, D4⋊C4, C4≀C2, C4×D7, D28, C7⋊D4, D4⋊C8, C8×D7, C8⋊D7, C56⋊C2, D56, D14⋊C4, Dic14⋊C4, D14⋊C8, C2.D56, C4.17D56
►On 224 points
Generators in S224
(1 196 151 63)(2 197 152 64)(3 198 153 65)(4 199 154 66)(5 200 155 67)(6 201 156 68)(7 202 157 69)(8 203 158 70)(9 204 159 71)(10 205 160 72)(11 206 161 73)(12 207 162 74)(13 208 163 75)(14 209 164 76)(15 210 165 77)(16 211 166 78)(17 212 167 79)(18 213 168 80)(19 214 113 81)(20 215 114 82)(21 216 115 83)(22 217 116 84)(23 218 117 85)(24 219 118 86)(25 220 119 87)(26 221 120 88)(27 222 121 89)(28 223 122 90)(29 224 123 91)(30 169 124 92)(31 170 125 93)(32 171 126 94)(33 172 127 95)(34 173 128 96)(35 174 129 97)(36 175 130 98)(37 176 131 99)(38 177 132 100)(39 178 133 101)(40 179 134 102)(41 180 135 103)(42 181 136 104)(43 182 137 105)(44 183 138 106)(45 184 139 107)(46 185 140 108)(47 186 141 109)(48 187 142 110)(49 188 143 111)(50 189 144 112)(51 190 145 57)(52 191 146 58)(53 192 147 59)(54 193 148 60)(55 194 149 61)(56 195 150 62)
(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)
(1 195 196 150 151 62 63 56)(2 149 197 61 152 55 64 194)(3 60 198 54 153 193 65 148)(4 53 199 192 154 147 66 59)(5 191 200 146 155 58 67 52)(6 145 201 57 156 51 68 190)(7 112 202 50 157 189 69 144)(8 49 203 188 158 143 70 111)(9 187 204 142 159 110 71 48)(10 141 205 109 160 47 72 186)(11 108 206 46 161 185 73 140)(12 45 207 184 162 139 74 107)(13 183 208 138 163 106 75 44)(14 137 209 105 164 43 76 182)(15 104 210 42 165 181 77 136)(16 41 211 180 166 135 78 103)(17 179 212 134 167 102 79 40)(18 133 213 101 168 39 80 178)(19 100 214 38 113 177 81 132)(20 37 215 176 114 131 82 99)(21 175 216 130 115 98 83 36)(22 129 217 97 116 35 84 174)(23 96 218 34 117 173 85 128)(24 33 219 172 118 127 86 95)(25 171 220 126 119 94 87 32)(26 125 221 93 120 31 88 170)(27 92 222 30 121 169 89 124)(28 29 223 224 122 123 90 91)
G:=sub<Sym(224)| (1,196,151,63)(2,197,152,64)(3,198,153,65)(4,199,154,66)(5,200,155,67)(6,201,156,68)(7,202,157,69)(8,203,158,70)(9,204,159,71)(10,205,160,72)(11,206,161,73)(12,207,162,74)(13,208,163,75)(14,209,164,76)(15,210,165,77)(16,211,166,78)(17,212,167,79)(18,213,168,80)(19,214,113,81)(20,215,114,82)(21,216,115,83)(22,217,116,84)(23,218,117,85)(24,219,118,86)(25,220,119,87)(26,221,120,88)(27,222,121,89)(28,223,122,90)(29,224,123,91)(30,169,124,92)(31,170,125,93)(32,171,126,94)(33,172,127,95)(34,173,128,96)(35,174,129,97)(36,175,130,98)(37,176,131,99)(38,177,132,100)(39,178,133,101)(40,179,134,102)(41,180,135,103)(42,181,136,104)(43,182,137,105)(44,183,138,106)(45,184,139,107)(46,185,140,108)(47,186,141,109)(48,187,142,110)(49,188,143,111)(50,189,144,112)(51,190,145,57)(52,191,146,58)(53,192,147,59)(54,193,148,60)(55,194,149,61)(56,195,150,62), (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), (1,195,196,150,151,62,63,56)(2,149,197,61,152,55,64,194)(3,60,198,54,153,193,65,148)(4,53,199,192,154,147,66,59)(5,191,200,146,155,58,67,52)(6,145,201,57,156,51,68,190)(7,112,202,50,157,189,69,144)(8,49,203,188,158,143,70,111)(9,187,204,142,159,110,71,48)(10,141,205,109,160,47,72,186)(11,108,206,46,161,185,73,140)(12,45,207,184,162,139,74,107)(13,183,208,138,163,106,75,44)(14,137,209,105,164,43,76,182)(15,104,210,42,165,181,77,136)(16,41,211,180,166,135,78,103)(17,179,212,134,167,102,79,40)(18,133,213,101,168,39,80,178)(19,100,214,38,113,177,81,132)(20,37,215,176,114,131,82,99)(21,175,216,130,115,98,83,36)(22,129,217,97,116,35,84,174)(23,96,218,34,117,173,85,128)(24,33,219,172,118,127,86,95)(25,171,220,126,119,94,87,32)(26,125,221,93,120,31,88,170)(27,92,222,30,121,169,89,124)(28,29,223,224,122,123,90,91)>;
G:=Group( (1,196,151,63)(2,197,152,64)(3,198,153,65)(4,199,154,66)(5,200,155,67)(6,201,156,68)(7,202,157,69)(8,203,158,70)(9,204,159,71)(10,205,160,72)(11,206,161,73)(12,207,162,74)(13,208,163,75)(14,209,164,76)(15,210,165,77)(16,211,166,78)(17,212,167,79)(18,213,168,80)(19,214,113,81)(20,215,114,82)(21,216,115,83)(22,217,116,84)(23,218,117,85)(24,219,118,86)(25,220,119,87)(26,221,120,88)(27,222,121,89)(28,223,122,90)(29,224,123,91)(30,169,124,92)(31,170,125,93)(32,171,126,94)(33,172,127,95)(34,173,128,96)(35,174,129,97)(36,175,130,98)(37,176,131,99)(38,177,132,100)(39,178,133,101)(40,179,134,102)(41,180,135,103)(42,181,136,104)(43,182,137,105)(44,183,138,106)(45,184,139,107)(46,185,140,108)(47,186,141,109)(48,187,142,110)(49,188,143,111)(50,189,144,112)(51,190,145,57)(52,191,146,58)(53,192,147,59)(54,193,148,60)(55,194,149,61)(56,195,150,62), (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), (1,195,196,150,151,62,63,56)(2,149,197,61,152,55,64,194)(3,60,198,54,153,193,65,148)(4,53,199,192,154,147,66,59)(5,191,200,146,155,58,67,52)(6,145,201,57,156,51,68,190)(7,112,202,50,157,189,69,144)(8,49,203,188,158,143,70,111)(9,187,204,142,159,110,71,48)(10,141,205,109,160,47,72,186)(11,108,206,46,161,185,73,140)(12,45,207,184,162,139,74,107)(13,183,208,138,163,106,75,44)(14,137,209,105,164,43,76,182)(15,104,210,42,165,181,77,136)(16,41,211,180,166,135,78,103)(17,179,212,134,167,102,79,40)(18,133,213,101,168,39,80,178)(19,100,214,38,113,177,81,132)(20,37,215,176,114,131,82,99)(21,175,216,130,115,98,83,36)(22,129,217,97,116,35,84,174)(23,96,218,34,117,173,85,128)(24,33,219,172,118,127,86,95)(25,171,220,126,119,94,87,32)(26,125,221,93,120,31,88,170)(27,92,222,30,121,169,89,124)(28,29,223,224,122,123,90,91) );
G=PermutationGroup([[(1,196,151,63),(2,197,152,64),(3,198,153,65),(4,199,154,66),(5,200,155,67),(6,201,156,68),(7,202,157,69),(8,203,158,70),(9,204,159,71),(10,205,160,72),(11,206,161,73),(12,207,162,74),(13,208,163,75),(14,209,164,76),(15,210,165,77),(16,211,166,78),(17,212,167,79),(18,213,168,80),(19,214,113,81),(20,215,114,82),(21,216,115,83),(22,217,116,84),(23,218,117,85),(24,219,118,86),(25,220,119,87),(26,221,120,88),(27,222,121,89),(28,223,122,90),(29,224,123,91),(30,169,124,92),(31,170,125,93),(32,171,126,94),(33,172,127,95),(34,173,128,96),(35,174,129,97),(36,175,130,98),(37,176,131,99),(38,177,132,100),(39,178,133,101),(40,179,134,102),(41,180,135,103),(42,181,136,104),(43,182,137,105),(44,183,138,106),(45,184,139,107),(46,185,140,108),(47,186,141,109),(48,187,142,110),(49,188,143,111),(50,189,144,112),(51,190,145,57),(52,191,146,58),(53,192,147,59),(54,193,148,60),(55,194,149,61),(56,195,150,62)], [(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)], [(1,195,196,150,151,62,63,56),(2,149,197,61,152,55,64,194),(3,60,198,54,153,193,65,148),(4,53,199,192,154,147,66,59),(5,191,200,146,155,58,67,52),(6,145,201,57,156,51,68,190),(7,112,202,50,157,189,69,144),(8,49,203,188,158,143,70,111),(9,187,204,142,159,110,71,48),(10,141,205,109,160,47,72,186),(11,108,206,46,161,185,73,140),(12,45,207,184,162,139,74,107),(13,183,208,138,163,106,75,44),(14,137,209,105,164,43,76,182),(15,104,210,42,165,181,77,136),(16,41,211,180,166,135,78,103),(17,179,212,134,167,102,79,40),(18,133,213,101,168,39,80,178),(19,100,214,38,113,177,81,132),(20,37,215,176,114,131,82,99),(21,175,216,130,115,98,83,36),(22,129,217,97,116,35,84,174),(23,96,218,34,117,173,85,128),(24,33,219,172,118,127,86,95),(25,171,220,126,119,94,87,32),(26,125,221,93,120,31,88,170),(27,92,222,30,121,169,89,124),(28,29,223,224,122,123,90,91)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D7 | M4(2) | D8 | SD16 | D14 | C4≀C2 | C4×D7 | D28 | C7⋊D4 | C8×D7 | C8⋊D7 | C56⋊C2 | D56 | Dic14⋊C4 |
| kernel | C4.17D56 | C28⋊C8 | C4×C56 | C4×D28 | C4⋊Dic7 | C2×D28 | D28 | C2×C28 | C4×C8 | C28 | C28 | C28 | C42 | C14 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 3 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 24 |
Matrix representation of C4.17D56 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 15 | 0 |
| 0 | 0 | 15 |
| 95 | 0 | 0 |
| 0 | 29 | 80 |
| 0 | 33 | 37 |
| 18 | 0 | 0 |
| 0 | 37 | 65 |
| 0 | 80 | 76 |
G:=sub<GL(3,GF(113))| [98,0,0,0,15,0,0,0,15],[95,0,0,0,29,33,0,80,37],[18,0,0,0,37,80,0,65,76] >;
C4.17D56 in GAP, Magma, Sage, TeX
C_4._{17}D_{56} % in TeX
G:=Group("C4.17D56"); // GroupNames label
G:=SmallGroup(448,16);
// by ID
G=gap.SmallGroup(448,16);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,100,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^4=b^56=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations