direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic27, C54⋊C4, C2.2D54, C22.D27, C6.11D18, C18.11D6, C6.2Dic9, C54.4C22, C18.2Dic3, C27⋊2(C2×C4), (C2×C54).C2, (C2×C6).2D9, C9.(C2×Dic3), C3.(C2×Dic9), (C2×C18).2S3, SmallGroup(216,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C2×Dic27 |
Generators and relations for C2×Dic27
G = < a,b,c | a2=b54=1, c2=b27, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic27
►Regular action on 216 points
Generators in S216
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 55)(39 56)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(109 194)(110 195)(111 196)(112 197)(113 198)(114 199)(115 200)(116 201)(117 202)(118 203)(119 204)(120 205)(121 206)(122 207)(123 208)(124 209)(125 210)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 169)(139 170)(140 171)(141 172)(142 173)(143 174)(144 175)(145 176)(146 177)(147 178)(148 179)(149 180)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)
(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)
(1 171 28 198)(2 170 29 197)(3 169 30 196)(4 168 31 195)(5 167 32 194)(6 166 33 193)(7 165 34 192)(8 164 35 191)(9 163 36 190)(10 216 37 189)(11 215 38 188)(12 214 39 187)(13 213 40 186)(14 212 41 185)(15 211 42 184)(16 210 43 183)(17 209 44 182)(18 208 45 181)(19 207 46 180)(20 206 47 179)(21 205 48 178)(22 204 49 177)(23 203 50 176)(24 202 51 175)(25 201 52 174)(26 200 53 173)(27 199 54 172)(55 157 82 130)(56 156 83 129)(57 155 84 128)(58 154 85 127)(59 153 86 126)(60 152 87 125)(61 151 88 124)(62 150 89 123)(63 149 90 122)(64 148 91 121)(65 147 92 120)(66 146 93 119)(67 145 94 118)(68 144 95 117)(69 143 96 116)(70 142 97 115)(71 141 98 114)(72 140 99 113)(73 139 100 112)(74 138 101 111)(75 137 102 110)(76 136 103 109)(77 135 104 162)(78 134 105 161)(79 133 106 160)(80 132 107 159)(81 131 108 158)
G:=sub<Sym(216)| (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193), (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), (1,171,28,198)(2,170,29,197)(3,169,30,196)(4,168,31,195)(5,167,32,194)(6,166,33,193)(7,165,34,192)(8,164,35,191)(9,163,36,190)(10,216,37,189)(11,215,38,188)(12,214,39,187)(13,213,40,186)(14,212,41,185)(15,211,42,184)(16,210,43,183)(17,209,44,182)(18,208,45,181)(19,207,46,180)(20,206,47,179)(21,205,48,178)(22,204,49,177)(23,203,50,176)(24,202,51,175)(25,201,52,174)(26,200,53,173)(27,199,54,172)(55,157,82,130)(56,156,83,129)(57,155,84,128)(58,154,85,127)(59,153,86,126)(60,152,87,125)(61,151,88,124)(62,150,89,123)(63,149,90,122)(64,148,91,121)(65,147,92,120)(66,146,93,119)(67,145,94,118)(68,144,95,117)(69,143,96,116)(70,142,97,115)(71,141,98,114)(72,140,99,113)(73,139,100,112)(74,138,101,111)(75,137,102,110)(76,136,103,109)(77,135,104,162)(78,134,105,161)(79,133,106,160)(80,132,107,159)(81,131,108,158)>;
G:=Group( (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,55)(39,56)(40,57)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,65)(49,66)(50,67)(51,68)(52,69)(53,70)(54,71)(109,194)(110,195)(111,196)(112,197)(113,198)(114,199)(115,200)(116,201)(117,202)(118,203)(119,204)(120,205)(121,206)(122,207)(123,208)(124,209)(125,210)(126,211)(127,212)(128,213)(129,214)(130,215)(131,216)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,169)(139,170)(140,171)(141,172)(142,173)(143,174)(144,175)(145,176)(146,177)(147,178)(148,179)(149,180)(150,181)(151,182)(152,183)(153,184)(154,185)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193), (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), (1,171,28,198)(2,170,29,197)(3,169,30,196)(4,168,31,195)(5,167,32,194)(6,166,33,193)(7,165,34,192)(8,164,35,191)(9,163,36,190)(10,216,37,189)(11,215,38,188)(12,214,39,187)(13,213,40,186)(14,212,41,185)(15,211,42,184)(16,210,43,183)(17,209,44,182)(18,208,45,181)(19,207,46,180)(20,206,47,179)(21,205,48,178)(22,204,49,177)(23,203,50,176)(24,202,51,175)(25,201,52,174)(26,200,53,173)(27,199,54,172)(55,157,82,130)(56,156,83,129)(57,155,84,128)(58,154,85,127)(59,153,86,126)(60,152,87,125)(61,151,88,124)(62,150,89,123)(63,149,90,122)(64,148,91,121)(65,147,92,120)(66,146,93,119)(67,145,94,118)(68,144,95,117)(69,143,96,116)(70,142,97,115)(71,141,98,114)(72,140,99,113)(73,139,100,112)(74,138,101,111)(75,137,102,110)(76,136,103,109)(77,135,104,162)(78,134,105,161)(79,133,106,160)(80,132,107,159)(81,131,108,158) );
G=PermutationGroup([[(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,55),(39,56),(40,57),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,65),(49,66),(50,67),(51,68),(52,69),(53,70),(54,71),(109,194),(110,195),(111,196),(112,197),(113,198),(114,199),(115,200),(116,201),(117,202),(118,203),(119,204),(120,205),(121,206),(122,207),(123,208),(124,209),(125,210),(126,211),(127,212),(128,213),(129,214),(130,215),(131,216),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,169),(139,170),(140,171),(141,172),(142,173),(143,174),(144,175),(145,176),(146,177),(147,178),(148,179),(149,180),(150,181),(151,182),(152,183),(153,184),(154,185),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193)], [(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)], [(1,171,28,198),(2,170,29,197),(3,169,30,196),(4,168,31,195),(5,167,32,194),(6,166,33,193),(7,165,34,192),(8,164,35,191),(9,163,36,190),(10,216,37,189),(11,215,38,188),(12,214,39,187),(13,213,40,186),(14,212,41,185),(15,211,42,184),(16,210,43,183),(17,209,44,182),(18,208,45,181),(19,207,46,180),(20,206,47,179),(21,205,48,178),(22,204,49,177),(23,203,50,176),(24,202,51,175),(25,201,52,174),(26,200,53,173),(27,199,54,172),(55,157,82,130),(56,156,83,129),(57,155,84,128),(58,154,85,127),(59,153,86,126),(60,152,87,125),(61,151,88,124),(62,150,89,123),(63,149,90,122),(64,148,91,121),(65,147,92,120),(66,146,93,119),(67,145,94,118),(68,144,95,117),(69,143,96,116),(70,142,97,115),(71,141,98,114),(72,140,99,113),(73,139,100,112),(74,138,101,111),(75,137,102,110),(76,136,103,109),(77,135,104,162),(78,134,105,161),(79,133,106,160),(80,132,107,159),(81,131,108,158)]])
C2×Dic27 is a maximal subgroup of
Dic27⋊C4 C4⋊Dic27 D54⋊C4 C54.D4 C2×C4×D27 D4⋊2D27
C2×Dic27 is a maximal quotient of C4.Dic27 C4⋊Dic27 C54.D4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 18A | ··· | 18I | 27A | ··· | 27I | 54A | ··· | 54AA |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 2 | 27 | 27 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | Dic3 | D6 | D9 | Dic9 | D18 | D27 | Dic27 | D54 |
| kernel | C2×Dic27 | Dic27 | C2×C54 | C54 | C2×C18 | C18 | C18 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 3 | 6 | 3 | 9 | 18 | 9 |
Matrix representation of C2×Dic27 ►in GL4(𝔽109) generated by
| 108 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 108 | 108 | 0 | 0 |
| 0 | 0 | 22 | 58 |
| 0 | 0 | 51 | 80 |
| 63 | 93 | 0 | 0 |
| 30 | 46 | 0 | 0 |
| 0 | 0 | 1 | 79 |
| 0 | 0 | 80 | 108 |
G:=sub<GL(4,GF(109))| [108,0,0,0,0,108,0,0,0,0,1,0,0,0,0,1],[0,108,0,0,1,108,0,0,0,0,22,51,0,0,58,80],[63,30,0,0,93,46,0,0,0,0,1,80,0,0,79,108] >;
C2×Dic27 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{27} % in TeX
G:=Group("C2xDic27"); // GroupNames label
G:=SmallGroup(216,7);
// by ID
G=gap.SmallGroup(216,7);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,963,381,3604,208,5189]);
// Polycyclic
G:=Group<a,b,c|a^2=b^54=1,c^2=b^27,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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