metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic27, C108.1C4, C36.63D6, C4.16D54, C27⋊2M4(2), C12.63D18, C36.1Dic3, C12.1Dic9, C22.Dic27, C108.16C22, C27⋊C8⋊5C2, (C2×C54).3C4, C54.7(C2×C4), (C2×C4).2D27, (C2×C36).10S3, (C2×C108).5C2, (C2×C12).10D9, (C2×C6).4Dic9, C6.7(C2×Dic9), C9.(C4.Dic3), C3.(C4.Dic9), C18.7(C2×Dic3), C2.3(C2×Dic27), (C2×C18).4Dic3, SmallGroup(432,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic27
G = < a,b,c | a4=1, b54=a2, c2=a2b27, ab=ba, cac-1=a-1, cbc-1=b53 >
Smallest permutation representation of C4.Dic27
►On 216 points
Generators in S216
(1 82 55 28)(2 83 56 29)(3 84 57 30)(4 85 58 31)(5 86 59 32)(6 87 60 33)(7 88 61 34)(8 89 62 35)(9 90 63 36)(10 91 64 37)(11 92 65 38)(12 93 66 39)(13 94 67 40)(14 95 68 41)(15 96 69 42)(16 97 70 43)(17 98 71 44)(18 99 72 45)(19 100 73 46)(20 101 74 47)(21 102 75 48)(22 103 76 49)(23 104 77 50)(24 105 78 51)(25 106 79 52)(26 107 80 53)(27 108 81 54)(109 136 163 190)(110 137 164 191)(111 138 165 192)(112 139 166 193)(113 140 167 194)(114 141 168 195)(115 142 169 196)(116 143 170 197)(117 144 171 198)(118 145 172 199)(119 146 173 200)(120 147 174 201)(121 148 175 202)(122 149 176 203)(123 150 177 204)(124 151 178 205)(125 152 179 206)(126 153 180 207)(127 154 181 208)(128 155 182 209)(129 156 183 210)(130 157 184 211)(131 158 185 212)(132 159 186 213)(133 160 187 214)(134 161 188 215)(135 162 189 216)
(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)
(1 159 82 132 55 213 28 186)(2 212 83 185 56 158 29 131)(3 157 84 130 57 211 30 184)(4 210 85 183 58 156 31 129)(5 155 86 128 59 209 32 182)(6 208 87 181 60 154 33 127)(7 153 88 126 61 207 34 180)(8 206 89 179 62 152 35 125)(9 151 90 124 63 205 36 178)(10 204 91 177 64 150 37 123)(11 149 92 122 65 203 38 176)(12 202 93 175 66 148 39 121)(13 147 94 120 67 201 40 174)(14 200 95 173 68 146 41 119)(15 145 96 118 69 199 42 172)(16 198 97 171 70 144 43 117)(17 143 98 116 71 197 44 170)(18 196 99 169 72 142 45 115)(19 141 100 114 73 195 46 168)(20 194 101 167 74 140 47 113)(21 139 102 112 75 193 48 166)(22 192 103 165 76 138 49 111)(23 137 104 110 77 191 50 164)(24 190 105 163 78 136 51 109)(25 135 106 216 79 189 52 162)(26 188 107 161 80 134 53 215)(27 133 108 214 81 187 54 160)
G:=sub<Sym(216)| (1,82,55,28)(2,83,56,29)(3,84,57,30)(4,85,58,31)(5,86,59,32)(6,87,60,33)(7,88,61,34)(8,89,62,35)(9,90,63,36)(10,91,64,37)(11,92,65,38)(12,93,66,39)(13,94,67,40)(14,95,68,41)(15,96,69,42)(16,97,70,43)(17,98,71,44)(18,99,72,45)(19,100,73,46)(20,101,74,47)(21,102,75,48)(22,103,76,49)(23,104,77,50)(24,105,78,51)(25,106,79,52)(26,107,80,53)(27,108,81,54)(109,136,163,190)(110,137,164,191)(111,138,165,192)(112,139,166,193)(113,140,167,194)(114,141,168,195)(115,142,169,196)(116,143,170,197)(117,144,171,198)(118,145,172,199)(119,146,173,200)(120,147,174,201)(121,148,175,202)(122,149,176,203)(123,150,177,204)(124,151,178,205)(125,152,179,206)(126,153,180,207)(127,154,181,208)(128,155,182,209)(129,156,183,210)(130,157,184,211)(131,158,185,212)(132,159,186,213)(133,160,187,214)(134,161,188,215)(135,162,189,216), (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), (1,159,82,132,55,213,28,186)(2,212,83,185,56,158,29,131)(3,157,84,130,57,211,30,184)(4,210,85,183,58,156,31,129)(5,155,86,128,59,209,32,182)(6,208,87,181,60,154,33,127)(7,153,88,126,61,207,34,180)(8,206,89,179,62,152,35,125)(9,151,90,124,63,205,36,178)(10,204,91,177,64,150,37,123)(11,149,92,122,65,203,38,176)(12,202,93,175,66,148,39,121)(13,147,94,120,67,201,40,174)(14,200,95,173,68,146,41,119)(15,145,96,118,69,199,42,172)(16,198,97,171,70,144,43,117)(17,143,98,116,71,197,44,170)(18,196,99,169,72,142,45,115)(19,141,100,114,73,195,46,168)(20,194,101,167,74,140,47,113)(21,139,102,112,75,193,48,166)(22,192,103,165,76,138,49,111)(23,137,104,110,77,191,50,164)(24,190,105,163,78,136,51,109)(25,135,106,216,79,189,52,162)(26,188,107,161,80,134,53,215)(27,133,108,214,81,187,54,160)>;
G:=Group( (1,82,55,28)(2,83,56,29)(3,84,57,30)(4,85,58,31)(5,86,59,32)(6,87,60,33)(7,88,61,34)(8,89,62,35)(9,90,63,36)(10,91,64,37)(11,92,65,38)(12,93,66,39)(13,94,67,40)(14,95,68,41)(15,96,69,42)(16,97,70,43)(17,98,71,44)(18,99,72,45)(19,100,73,46)(20,101,74,47)(21,102,75,48)(22,103,76,49)(23,104,77,50)(24,105,78,51)(25,106,79,52)(26,107,80,53)(27,108,81,54)(109,136,163,190)(110,137,164,191)(111,138,165,192)(112,139,166,193)(113,140,167,194)(114,141,168,195)(115,142,169,196)(116,143,170,197)(117,144,171,198)(118,145,172,199)(119,146,173,200)(120,147,174,201)(121,148,175,202)(122,149,176,203)(123,150,177,204)(124,151,178,205)(125,152,179,206)(126,153,180,207)(127,154,181,208)(128,155,182,209)(129,156,183,210)(130,157,184,211)(131,158,185,212)(132,159,186,213)(133,160,187,214)(134,161,188,215)(135,162,189,216), (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), (1,159,82,132,55,213,28,186)(2,212,83,185,56,158,29,131)(3,157,84,130,57,211,30,184)(4,210,85,183,58,156,31,129)(5,155,86,128,59,209,32,182)(6,208,87,181,60,154,33,127)(7,153,88,126,61,207,34,180)(8,206,89,179,62,152,35,125)(9,151,90,124,63,205,36,178)(10,204,91,177,64,150,37,123)(11,149,92,122,65,203,38,176)(12,202,93,175,66,148,39,121)(13,147,94,120,67,201,40,174)(14,200,95,173,68,146,41,119)(15,145,96,118,69,199,42,172)(16,198,97,171,70,144,43,117)(17,143,98,116,71,197,44,170)(18,196,99,169,72,142,45,115)(19,141,100,114,73,195,46,168)(20,194,101,167,74,140,47,113)(21,139,102,112,75,193,48,166)(22,192,103,165,76,138,49,111)(23,137,104,110,77,191,50,164)(24,190,105,163,78,136,51,109)(25,135,106,216,79,189,52,162)(26,188,107,161,80,134,53,215)(27,133,108,214,81,187,54,160) );
G=PermutationGroup([[(1,82,55,28),(2,83,56,29),(3,84,57,30),(4,85,58,31),(5,86,59,32),(6,87,60,33),(7,88,61,34),(8,89,62,35),(9,90,63,36),(10,91,64,37),(11,92,65,38),(12,93,66,39),(13,94,67,40),(14,95,68,41),(15,96,69,42),(16,97,70,43),(17,98,71,44),(18,99,72,45),(19,100,73,46),(20,101,74,47),(21,102,75,48),(22,103,76,49),(23,104,77,50),(24,105,78,51),(25,106,79,52),(26,107,80,53),(27,108,81,54),(109,136,163,190),(110,137,164,191),(111,138,165,192),(112,139,166,193),(113,140,167,194),(114,141,168,195),(115,142,169,196),(116,143,170,197),(117,144,171,198),(118,145,172,199),(119,146,173,200),(120,147,174,201),(121,148,175,202),(122,149,176,203),(123,150,177,204),(124,151,178,205),(125,152,179,206),(126,153,180,207),(127,154,181,208),(128,155,182,209),(129,156,183,210),(130,157,184,211),(131,158,185,212),(132,159,186,213),(133,160,187,214),(134,161,188,215),(135,162,189,216)], [(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)], [(1,159,82,132,55,213,28,186),(2,212,83,185,56,158,29,131),(3,157,84,130,57,211,30,184),(4,210,85,183,58,156,31,129),(5,155,86,128,59,209,32,182),(6,208,87,181,60,154,33,127),(7,153,88,126,61,207,34,180),(8,206,89,179,62,152,35,125),(9,151,90,124,63,205,36,178),(10,204,91,177,64,150,37,123),(11,149,92,122,65,203,38,176),(12,202,93,175,66,148,39,121),(13,147,94,120,67,201,40,174),(14,200,95,173,68,146,41,119),(15,145,96,118,69,199,42,172),(16,198,97,171,70,144,43,117),(17,143,98,116,71,197,44,170),(18,196,99,169,72,142,45,115),(19,141,100,114,73,195,46,168),(20,194,101,167,74,140,47,113),(21,139,102,112,75,193,48,166),(22,192,103,165,76,138,49,111),(23,137,104,110,77,191,50,164),(24,190,105,163,78,136,51,109),(25,135,106,216,79,189,52,162),(26,188,107,161,80,134,53,215),(27,133,108,214,81,187,54,160)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 27A | ··· | 27I | 36A | ··· | 36L | 54A | ··· | 54AA | 108A | ··· | 108AJ |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 54 | 54 | 54 | 54 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
114 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | D9 | Dic9 | D18 | Dic9 | C4.Dic3 | D27 | Dic27 | D54 | Dic27 | C4.Dic9 | C4.Dic27 |
| kernel | C4.Dic27 | C27⋊C8 | C2×C108 | C108 | C2×C54 | C2×C36 | C36 | C36 | C2×C18 | C27 | C2×C12 | C12 | C12 | C2×C6 | C9 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 9 | 9 | 9 | 9 | 12 | 36 |
Matrix representation of C4.Dic27 ►in GL2(𝔽433) generated by
| 179 | 0 |
| 63 | 254 |
| 123 | 0 |
| 317 | 88 |
| 347 | 310 |
| 115 | 86 |
G:=sub<GL(2,GF(433))| [179,63,0,254],[123,317,0,88],[347,115,310,86] >;
C4.Dic27 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{27} % in TeX
G:=Group("C4.Dic27"); // GroupNames label
G:=SmallGroup(432,10);
// by ID
G=gap.SmallGroup(432,10);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^54=a^2,c^2=a^2*b^27,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^53>;
// generators/relations
Export