metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D54⋊C4, C6.5D36, C54.6D4, C18.5D12, C2.2D108, C22.6D54, (C2×C4)⋊1D27, C9.(D6⋊C4), C6.9(C4×D9), (C2×C108)⋊1C2, C3.(D18⋊C4), (C2×C36).2S3, C54.5(C2×C4), (C2×C12).2D9, C2.5(C4×D27), C18.10(C4×S3), C27⋊1(C22⋊C4), (C2×C6).27D18, (C2×C18).27D6, (C22×D27).C2, (C2×Dic27)⋊1C2, C2.2(C27⋊D4), C6.15(C9⋊D4), (C2×C54).6C22, C18.15(C3⋊D4), SmallGroup(432,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D54⋊C4
G = < a,b,c | a54=b2=c4=1, bab=a-1, ac=ca, cbc-1=a27b >
Subgroups: 632 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C22⋊C4, D9, C18, C2×Dic3, C2×C12, C22×S3, C27, Dic9, C36, D18, C2×C18, D6⋊C4, D27, C54, C2×Dic9, C2×C36, C22×D9, Dic27, C108, D54, D54, C2×C54, D18⋊C4, C2×Dic27, C2×C108, C22×D27, D54⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, D18, D6⋊C4, D27, C4×D9, D36, C9⋊D4, D54, D18⋊C4, C4×D27, D108, C27⋊D4, D54⋊C4
►On 216 points
Generators in S216
(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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(55 58)(56 57)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(109 153)(110 152)(111 151)(112 150)(113 149)(114 148)(115 147)(116 146)(117 145)(118 144)(119 143)(120 142)(121 141)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(154 162)(155 161)(156 160)(157 159)(163 185)(164 184)(165 183)(166 182)(167 181)(168 180)(169 179)(170 178)(171 177)(172 176)(173 175)(186 216)(187 215)(188 214)(189 213)(190 212)(191 211)(192 210)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)
(1 188 84 118)(2 189 85 119)(3 190 86 120)(4 191 87 121)(5 192 88 122)(6 193 89 123)(7 194 90 124)(8 195 91 125)(9 196 92 126)(10 197 93 127)(11 198 94 128)(12 199 95 129)(13 200 96 130)(14 201 97 131)(15 202 98 132)(16 203 99 133)(17 204 100 134)(18 205 101 135)(19 206 102 136)(20 207 103 137)(21 208 104 138)(22 209 105 139)(23 210 106 140)(24 211 107 141)(25 212 108 142)(26 213 55 143)(27 214 56 144)(28 215 57 145)(29 216 58 146)(30 163 59 147)(31 164 60 148)(32 165 61 149)(33 166 62 150)(34 167 63 151)(35 168 64 152)(36 169 65 153)(37 170 66 154)(38 171 67 155)(39 172 68 156)(40 173 69 157)(41 174 70 158)(42 175 71 159)(43 176 72 160)(44 177 73 161)(45 178 74 162)(46 179 75 109)(47 180 76 110)(48 181 77 111)(49 182 78 112)(50 183 79 113)(51 184 80 114)(52 185 81 115)(53 186 82 116)(54 187 83 117)
G:=sub<Sym(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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(55,58)(56,57)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(154,162)(155,161)(156,160)(157,159)(163,185)(164,184)(165,183)(166,182)(167,181)(168,180)(169,179)(170,178)(171,177)(172,176)(173,175)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202), (1,188,84,118)(2,189,85,119)(3,190,86,120)(4,191,87,121)(5,192,88,122)(6,193,89,123)(7,194,90,124)(8,195,91,125)(9,196,92,126)(10,197,93,127)(11,198,94,128)(12,199,95,129)(13,200,96,130)(14,201,97,131)(15,202,98,132)(16,203,99,133)(17,204,100,134)(18,205,101,135)(19,206,102,136)(20,207,103,137)(21,208,104,138)(22,209,105,139)(23,210,106,140)(24,211,107,141)(25,212,108,142)(26,213,55,143)(27,214,56,144)(28,215,57,145)(29,216,58,146)(30,163,59,147)(31,164,60,148)(32,165,61,149)(33,166,62,150)(34,167,63,151)(35,168,64,152)(36,169,65,153)(37,170,66,154)(38,171,67,155)(39,172,68,156)(40,173,69,157)(41,174,70,158)(42,175,71,159)(43,176,72,160)(44,177,73,161)(45,178,74,162)(46,179,75,109)(47,180,76,110)(48,181,77,111)(49,182,78,112)(50,183,79,113)(51,184,80,114)(52,185,81,115)(53,186,82,116)(54,187,83,117)>;
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(55,58)(56,57)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,84)(109,153)(110,152)(111,151)(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(154,162)(155,161)(156,160)(157,159)(163,185)(164,184)(165,183)(166,182)(167,181)(168,180)(169,179)(170,178)(171,177)(172,176)(173,175)(186,216)(187,215)(188,214)(189,213)(190,212)(191,211)(192,210)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202), (1,188,84,118)(2,189,85,119)(3,190,86,120)(4,191,87,121)(5,192,88,122)(6,193,89,123)(7,194,90,124)(8,195,91,125)(9,196,92,126)(10,197,93,127)(11,198,94,128)(12,199,95,129)(13,200,96,130)(14,201,97,131)(15,202,98,132)(16,203,99,133)(17,204,100,134)(18,205,101,135)(19,206,102,136)(20,207,103,137)(21,208,104,138)(22,209,105,139)(23,210,106,140)(24,211,107,141)(25,212,108,142)(26,213,55,143)(27,214,56,144)(28,215,57,145)(29,216,58,146)(30,163,59,147)(31,164,60,148)(32,165,61,149)(33,166,62,150)(34,167,63,151)(35,168,64,152)(36,169,65,153)(37,170,66,154)(38,171,67,155)(39,172,68,156)(40,173,69,157)(41,174,70,158)(42,175,71,159)(43,176,72,160)(44,177,73,161)(45,178,74,162)(46,179,75,109)(47,180,76,110)(48,181,77,111)(49,182,78,112)(50,183,79,113)(51,184,80,114)(52,185,81,115)(53,186,82,116)(54,187,83,117) );
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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(55,58),(56,57),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,84),(109,153),(110,152),(111,151),(112,150),(113,149),(114,148),(115,147),(116,146),(117,145),(118,144),(119,143),(120,142),(121,141),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(154,162),(155,161),(156,160),(157,159),(163,185),(164,184),(165,183),(166,182),(167,181),(168,180),(169,179),(170,178),(171,177),(172,176),(173,175),(186,216),(187,215),(188,214),(189,213),(190,212),(191,211),(192,210),(193,209),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202)], [(1,188,84,118),(2,189,85,119),(3,190,86,120),(4,191,87,121),(5,192,88,122),(6,193,89,123),(7,194,90,124),(8,195,91,125),(9,196,92,126),(10,197,93,127),(11,198,94,128),(12,199,95,129),(13,200,96,130),(14,201,97,131),(15,202,98,132),(16,203,99,133),(17,204,100,134),(18,205,101,135),(19,206,102,136),(20,207,103,137),(21,208,104,138),(22,209,105,139),(23,210,106,140),(24,211,107,141),(25,212,108,142),(26,213,55,143),(27,214,56,144),(28,215,57,145),(29,216,58,146),(30,163,59,147),(31,164,60,148),(32,165,61,149),(33,166,62,150),(34,167,63,151),(35,168,64,152),(36,169,65,153),(37,170,66,154),(38,171,67,155),(39,172,68,156),(40,173,69,157),(41,174,70,158),(42,175,71,159),(43,176,72,160),(44,177,73,161),(45,178,74,162),(46,179,75,109),(47,180,76,110),(48,181,77,111),(49,182,78,112),(50,183,79,113),(51,184,80,114),(52,185,81,115),(53,186,82,116),(54,187,83,117)]])
114 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 27A | ··· | 27I | 36A | ··· | 36L | 54A | ··· | 54AA | 108A | ··· | 108AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 54 | 54 | 2 | 2 | 2 | 54 | 54 | 2 | 2 | 2 | 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 | C2 | C4 | S3 | D4 | D6 | D9 | C4×S3 | D12 | C3⋊D4 | D18 | D27 | C4×D9 | D36 | C9⋊D4 | D54 | C4×D27 | D108 | C27⋊D4 |
| kernel | D54⋊C4 | C2×Dic27 | C2×C108 | C22×D27 | D54 | C2×C36 | C54 | C2×C18 | C2×C12 | C18 | C18 | C18 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 3 | 9 | 6 | 6 | 6 | 9 | 18 | 18 | 18 |
Matrix representation of D54⋊C4 ►in GL4(𝔽109) generated by
| 108 | 1 | 0 | 0 |
| 108 | 0 | 0 | 0 |
| 0 | 0 | 80 | 58 |
| 0 | 0 | 51 | 22 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 30 | 93 |
| 0 | 0 | 63 | 79 |
| 33 | 0 | 0 | 0 |
| 0 | 33 | 0 | 0 |
| 0 | 0 | 6 | 12 |
| 0 | 0 | 97 | 103 |
G:=sub<GL(4,GF(109))| [108,108,0,0,1,0,0,0,0,0,80,51,0,0,58,22],[0,1,0,0,1,0,0,0,0,0,30,63,0,0,93,79],[33,0,0,0,0,33,0,0,0,0,6,97,0,0,12,103] >;
D54⋊C4 in GAP, Magma, Sage, TeX
D_{54}\rtimes C_4 % in TeX
G:=Group("D54:C4"); // GroupNames label
G:=SmallGroup(432,14);
// by ID
G=gap.SmallGroup(432,14);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^54=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^27*b>;
// generators/relations