metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.8D28, C56.58D4, C42.263D14, (C4×C8)⋊9D7, (C4×C56)⋊14C2, (C2×D56).2C2, C4.32(C2×D28), (C2×C4).62D28, (C2×Dic28)⋊1C2, C14.4(C4○D8), (C2×C28).352D4, C28.275(C2×D4), (C2×C8).318D14, C4.D28⋊1C2, C7⋊1(C8.12D4), C14.5(C4⋊1D4), C2.7(C28⋊4D4), (C2×D28).4C22, C22.92(C2×D28), C2.7(D56⋊7C2), (C2×C28).725C23, (C4×C28).309C22, (C2×C56).390C22, (C2×Dic14).3C22, (C2×C56⋊C2)⋊8C2, (C2×C14).108(C2×D4), (C2×C4).668(C22×D7), SmallGroup(448,230)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.8D28
G = < a,b,c | a8=b28=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 900 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C56, Dic14, D28, C2×Dic7, C2×C28, C2×C28, C22×D7, C8.12D4, C56⋊C2, D56, Dic28, D14⋊C4, C4×C28, C2×C56, C2×Dic14, C2×D28, C4×C56, C4.D28, C2×C56⋊C2, C2×D56, C2×Dic28, C8.8D28
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C4○D8, D28, C22×D7, C8.12D4, C2×D28, C28⋊4D4, D56⋊7C2, C8.8D28
►On 224 points
Generators in S224
(1 214 94 37 68 185 168 129)(2 215 95 38 69 186 141 130)(3 216 96 39 70 187 142 131)(4 217 97 40 71 188 143 132)(5 218 98 41 72 189 144 133)(6 219 99 42 73 190 145 134)(7 220 100 43 74 191 146 135)(8 221 101 44 75 192 147 136)(9 222 102 45 76 193 148 137)(10 223 103 46 77 194 149 138)(11 224 104 47 78 195 150 139)(12 197 105 48 79 196 151 140)(13 198 106 49 80 169 152 113)(14 199 107 50 81 170 153 114)(15 200 108 51 82 171 154 115)(16 201 109 52 83 172 155 116)(17 202 110 53 84 173 156 117)(18 203 111 54 57 174 157 118)(19 204 112 55 58 175 158 119)(20 205 85 56 59 176 159 120)(21 206 86 29 60 177 160 121)(22 207 87 30 61 178 161 122)(23 208 88 31 62 179 162 123)(24 209 89 32 63 180 163 124)(25 210 90 33 64 181 164 125)(26 211 91 34 65 182 165 126)(27 212 92 35 66 183 166 127)(28 213 93 36 67 184 167 128)
(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 36 68 128)(2 127 69 35)(3 34 70 126)(4 125 71 33)(5 32 72 124)(6 123 73 31)(7 30 74 122)(8 121 75 29)(9 56 76 120)(10 119 77 55)(11 54 78 118)(12 117 79 53)(13 52 80 116)(14 115 81 51)(15 50 82 114)(16 113 83 49)(17 48 84 140)(18 139 57 47)(19 46 58 138)(20 137 59 45)(21 44 60 136)(22 135 61 43)(23 42 62 134)(24 133 63 41)(25 40 64 132)(26 131 65 39)(27 38 66 130)(28 129 67 37)(85 193 159 222)(86 221 160 192)(87 191 161 220)(88 219 162 190)(89 189 163 218)(90 217 164 188)(91 187 165 216)(92 215 166 186)(93 185 167 214)(94 213 168 184)(95 183 141 212)(96 211 142 182)(97 181 143 210)(98 209 144 180)(99 179 145 208)(100 207 146 178)(101 177 147 206)(102 205 148 176)(103 175 149 204)(104 203 150 174)(105 173 151 202)(106 201 152 172)(107 171 153 200)(108 199 154 170)(109 169 155 198)(110 197 156 196)(111 195 157 224)(112 223 158 194)
G:=sub<Sym(224)| (1,214,94,37,68,185,168,129)(2,215,95,38,69,186,141,130)(3,216,96,39,70,187,142,131)(4,217,97,40,71,188,143,132)(5,218,98,41,72,189,144,133)(6,219,99,42,73,190,145,134)(7,220,100,43,74,191,146,135)(8,221,101,44,75,192,147,136)(9,222,102,45,76,193,148,137)(10,223,103,46,77,194,149,138)(11,224,104,47,78,195,150,139)(12,197,105,48,79,196,151,140)(13,198,106,49,80,169,152,113)(14,199,107,50,81,170,153,114)(15,200,108,51,82,171,154,115)(16,201,109,52,83,172,155,116)(17,202,110,53,84,173,156,117)(18,203,111,54,57,174,157,118)(19,204,112,55,58,175,158,119)(20,205,85,56,59,176,159,120)(21,206,86,29,60,177,160,121)(22,207,87,30,61,178,161,122)(23,208,88,31,62,179,162,123)(24,209,89,32,63,180,163,124)(25,210,90,33,64,181,164,125)(26,211,91,34,65,182,165,126)(27,212,92,35,66,183,166,127)(28,213,93,36,67,184,167,128), (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), (1,36,68,128)(2,127,69,35)(3,34,70,126)(4,125,71,33)(5,32,72,124)(6,123,73,31)(7,30,74,122)(8,121,75,29)(9,56,76,120)(10,119,77,55)(11,54,78,118)(12,117,79,53)(13,52,80,116)(14,115,81,51)(15,50,82,114)(16,113,83,49)(17,48,84,140)(18,139,57,47)(19,46,58,138)(20,137,59,45)(21,44,60,136)(22,135,61,43)(23,42,62,134)(24,133,63,41)(25,40,64,132)(26,131,65,39)(27,38,66,130)(28,129,67,37)(85,193,159,222)(86,221,160,192)(87,191,161,220)(88,219,162,190)(89,189,163,218)(90,217,164,188)(91,187,165,216)(92,215,166,186)(93,185,167,214)(94,213,168,184)(95,183,141,212)(96,211,142,182)(97,181,143,210)(98,209,144,180)(99,179,145,208)(100,207,146,178)(101,177,147,206)(102,205,148,176)(103,175,149,204)(104,203,150,174)(105,173,151,202)(106,201,152,172)(107,171,153,200)(108,199,154,170)(109,169,155,198)(110,197,156,196)(111,195,157,224)(112,223,158,194)>;
G:=Group( (1,214,94,37,68,185,168,129)(2,215,95,38,69,186,141,130)(3,216,96,39,70,187,142,131)(4,217,97,40,71,188,143,132)(5,218,98,41,72,189,144,133)(6,219,99,42,73,190,145,134)(7,220,100,43,74,191,146,135)(8,221,101,44,75,192,147,136)(9,222,102,45,76,193,148,137)(10,223,103,46,77,194,149,138)(11,224,104,47,78,195,150,139)(12,197,105,48,79,196,151,140)(13,198,106,49,80,169,152,113)(14,199,107,50,81,170,153,114)(15,200,108,51,82,171,154,115)(16,201,109,52,83,172,155,116)(17,202,110,53,84,173,156,117)(18,203,111,54,57,174,157,118)(19,204,112,55,58,175,158,119)(20,205,85,56,59,176,159,120)(21,206,86,29,60,177,160,121)(22,207,87,30,61,178,161,122)(23,208,88,31,62,179,162,123)(24,209,89,32,63,180,163,124)(25,210,90,33,64,181,164,125)(26,211,91,34,65,182,165,126)(27,212,92,35,66,183,166,127)(28,213,93,36,67,184,167,128), (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), (1,36,68,128)(2,127,69,35)(3,34,70,126)(4,125,71,33)(5,32,72,124)(6,123,73,31)(7,30,74,122)(8,121,75,29)(9,56,76,120)(10,119,77,55)(11,54,78,118)(12,117,79,53)(13,52,80,116)(14,115,81,51)(15,50,82,114)(16,113,83,49)(17,48,84,140)(18,139,57,47)(19,46,58,138)(20,137,59,45)(21,44,60,136)(22,135,61,43)(23,42,62,134)(24,133,63,41)(25,40,64,132)(26,131,65,39)(27,38,66,130)(28,129,67,37)(85,193,159,222)(86,221,160,192)(87,191,161,220)(88,219,162,190)(89,189,163,218)(90,217,164,188)(91,187,165,216)(92,215,166,186)(93,185,167,214)(94,213,168,184)(95,183,141,212)(96,211,142,182)(97,181,143,210)(98,209,144,180)(99,179,145,208)(100,207,146,178)(101,177,147,206)(102,205,148,176)(103,175,149,204)(104,203,150,174)(105,173,151,202)(106,201,152,172)(107,171,153,200)(108,199,154,170)(109,169,155,198)(110,197,156,196)(111,195,157,224)(112,223,158,194) );
G=PermutationGroup([[(1,214,94,37,68,185,168,129),(2,215,95,38,69,186,141,130),(3,216,96,39,70,187,142,131),(4,217,97,40,71,188,143,132),(5,218,98,41,72,189,144,133),(6,219,99,42,73,190,145,134),(7,220,100,43,74,191,146,135),(8,221,101,44,75,192,147,136),(9,222,102,45,76,193,148,137),(10,223,103,46,77,194,149,138),(11,224,104,47,78,195,150,139),(12,197,105,48,79,196,151,140),(13,198,106,49,80,169,152,113),(14,199,107,50,81,170,153,114),(15,200,108,51,82,171,154,115),(16,201,109,52,83,172,155,116),(17,202,110,53,84,173,156,117),(18,203,111,54,57,174,157,118),(19,204,112,55,58,175,158,119),(20,205,85,56,59,176,159,120),(21,206,86,29,60,177,160,121),(22,207,87,30,61,178,161,122),(23,208,88,31,62,179,162,123),(24,209,89,32,63,180,163,124),(25,210,90,33,64,181,164,125),(26,211,91,34,65,182,165,126),(27,212,92,35,66,183,166,127),(28,213,93,36,67,184,167,128)], [(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)], [(1,36,68,128),(2,127,69,35),(3,34,70,126),(4,125,71,33),(5,32,72,124),(6,123,73,31),(7,30,74,122),(8,121,75,29),(9,56,76,120),(10,119,77,55),(11,54,78,118),(12,117,79,53),(13,52,80,116),(14,115,81,51),(15,50,82,114),(16,113,83,49),(17,48,84,140),(18,139,57,47),(19,46,58,138),(20,137,59,45),(21,44,60,136),(22,135,61,43),(23,42,62,134),(24,133,63,41),(25,40,64,132),(26,131,65,39),(27,38,66,130),(28,129,67,37),(85,193,159,222),(86,221,160,192),(87,191,161,220),(88,219,162,190),(89,189,163,218),(90,217,164,188),(91,187,165,216),(92,215,166,186),(93,185,167,214),(94,213,168,184),(95,183,141,212),(96,211,142,182),(97,181,143,210),(98,209,144,180),(99,179,145,208),(100,207,146,178),(101,177,147,206),(102,205,148,176),(103,175,149,204),(104,203,150,174),(105,173,151,202),(106,201,152,172),(107,171,153,200),(108,199,154,170),(109,169,155,198),(110,197,156,196),(111,195,157,224),(112,223,158,194)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | ··· | 2 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C4○D8 | D28 | D28 | D56⋊7C2 |
| kernel | C8.8D28 | C4×C56 | C4.D28 | C2×C56⋊C2 | C2×D56 | C2×Dic28 | C56 | C2×C28 | C4×C8 | C42 | C2×C8 | C14 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 2 | 3 | 3 | 6 | 8 | 24 | 12 | 48 |
Matrix representation of C8.8D28 ►in GL4(𝔽113) generated by
| 21 | 70 | 0 | 0 |
| 93 | 41 | 0 | 0 |
| 0 | 0 | 21 | 91 |
| 0 | 0 | 2 | 41 |
| 85 | 82 | 0 | 0 |
| 25 | 60 | 0 | 0 |
| 0 | 0 | 22 | 92 |
| 0 | 0 | 43 | 0 |
| 42 | 78 | 0 | 0 |
| 2 | 71 | 0 | 0 |
| 0 | 0 | 71 | 43 |
| 0 | 0 | 1 | 42 |
G:=sub<GL(4,GF(113))| [21,93,0,0,70,41,0,0,0,0,21,2,0,0,91,41],[85,25,0,0,82,60,0,0,0,0,22,43,0,0,92,0],[42,2,0,0,78,71,0,0,0,0,71,1,0,0,43,42] >;
C8.8D28 in GAP, Magma, Sage, TeX
C_8._8D_{28} % in TeX
G:=Group("C8.8D28"); // GroupNames label
G:=SmallGroup(448,230);
// by ID
G=gap.SmallGroup(448,230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,58,1123,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=b^28=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations