direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C4○D4×C27, D4⋊2C54, Q8⋊3C54, C54.13C23, C108.21C22, (C2×C4)⋊3C54, (C2×C108)⋊7C2, (D4×C27)⋊5C2, C4.6(C2×C54), (Q8×C27)⋊5C2, (D4×C9).8C6, C22.(C2×C54), C36.45(C2×C6), (C2×C36).24C6, (C3×D4).6C18, (Q8×C9).14C6, (C2×C12).12C18, C12.22(C2×C18), (C2×C54).2C22, C2.3(C22×C54), (C3×Q8).10C18, C18.27(C22×C6), C6.13(C22×C18), C3.(C9×C4○D4), C9.(C3×C4○D4), (C2×C6).4(C2×C18), (C9×C4○D4).2C3, (C3×C4○D4).2C9, (C2×C18).21(C2×C6), SmallGroup(432,56)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D4×C27
G = < a,b,c,d | a27=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >
Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, D4, Q8, C9, C12, C12, C2×C6, C4○D4, C18, C18, C2×C12, C3×D4, C3×Q8, C27, C36, C36, C2×C18, C3×C4○D4, C54, C54, C2×C36, D4×C9, Q8×C9, C108, C108, C2×C54, C9×C4○D4, C2×C108, D4×C27, Q8×C27, C4○D4×C27
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C4○D4, C18, C22×C6, C27, C2×C18, C3×C4○D4, C54, C22×C18, C2×C54, C9×C4○D4, C22×C54, C4○D4×C27
►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)
(1 154 89 131)(2 155 90 132)(3 156 91 133)(4 157 92 134)(5 158 93 135)(6 159 94 109)(7 160 95 110)(8 161 96 111)(9 162 97 112)(10 136 98 113)(11 137 99 114)(12 138 100 115)(13 139 101 116)(14 140 102 117)(15 141 103 118)(16 142 104 119)(17 143 105 120)(18 144 106 121)(19 145 107 122)(20 146 108 123)(21 147 82 124)(22 148 83 125)(23 149 84 126)(24 150 85 127)(25 151 86 128)(26 152 87 129)(27 153 88 130)(28 81 216 182)(29 55 190 183)(30 56 191 184)(31 57 192 185)(32 58 193 186)(33 59 194 187)(34 60 195 188)(35 61 196 189)(36 62 197 163)(37 63 198 164)(38 64 199 165)(39 65 200 166)(40 66 201 167)(41 67 202 168)(42 68 203 169)(43 69 204 170)(44 70 205 171)(45 71 206 172)(46 72 207 173)(47 73 208 174)(48 74 209 175)(49 75 210 176)(50 76 211 177)(51 77 212 178)(52 78 213 179)(53 79 214 180)(54 80 215 181)
(1 74 89 175)(2 75 90 176)(3 76 91 177)(4 77 92 178)(5 78 93 179)(6 79 94 180)(7 80 95 181)(8 81 96 182)(9 55 97 183)(10 56 98 184)(11 57 99 185)(12 58 100 186)(13 59 101 187)(14 60 102 188)(15 61 103 189)(16 62 104 163)(17 63 105 164)(18 64 106 165)(19 65 107 166)(20 66 108 167)(21 67 82 168)(22 68 83 169)(23 69 84 170)(24 70 85 171)(25 71 86 172)(26 72 87 173)(27 73 88 174)(28 161 216 111)(29 162 190 112)(30 136 191 113)(31 137 192 114)(32 138 193 115)(33 139 194 116)(34 140 195 117)(35 141 196 118)(36 142 197 119)(37 143 198 120)(38 144 199 121)(39 145 200 122)(40 146 201 123)(41 147 202 124)(42 148 203 125)(43 149 204 126)(44 150 205 127)(45 151 206 128)(46 152 207 129)(47 153 208 130)(48 154 209 131)(49 155 210 132)(50 156 211 133)(51 157 212 134)(52 158 213 135)(53 159 214 109)(54 160 215 110)
(28 216)(29 190)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 197)(37 198)(38 199)(39 200)(40 201)(41 202)(42 203)(43 204)(44 205)(45 206)(46 207)(47 208)(48 209)(49 210)(50 211)(51 212)(52 213)(53 214)(54 215)(55 183)(56 184)(57 185)(58 186)(59 187)(60 188)(61 189)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 169)(69 170)(70 171)(71 172)(72 173)(73 174)(74 175)(75 176)(76 177)(77 178)(78 179)(79 180)(80 181)(81 182)
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), (1,154,89,131)(2,155,90,132)(3,156,91,133)(4,157,92,134)(5,158,93,135)(6,159,94,109)(7,160,95,110)(8,161,96,111)(9,162,97,112)(10,136,98,113)(11,137,99,114)(12,138,100,115)(13,139,101,116)(14,140,102,117)(15,141,103,118)(16,142,104,119)(17,143,105,120)(18,144,106,121)(19,145,107,122)(20,146,108,123)(21,147,82,124)(22,148,83,125)(23,149,84,126)(24,150,85,127)(25,151,86,128)(26,152,87,129)(27,153,88,130)(28,81,216,182)(29,55,190,183)(30,56,191,184)(31,57,192,185)(32,58,193,186)(33,59,194,187)(34,60,195,188)(35,61,196,189)(36,62,197,163)(37,63,198,164)(38,64,199,165)(39,65,200,166)(40,66,201,167)(41,67,202,168)(42,68,203,169)(43,69,204,170)(44,70,205,171)(45,71,206,172)(46,72,207,173)(47,73,208,174)(48,74,209,175)(49,75,210,176)(50,76,211,177)(51,77,212,178)(52,78,213,179)(53,79,214,180)(54,80,215,181), (1,74,89,175)(2,75,90,176)(3,76,91,177)(4,77,92,178)(5,78,93,179)(6,79,94,180)(7,80,95,181)(8,81,96,182)(9,55,97,183)(10,56,98,184)(11,57,99,185)(12,58,100,186)(13,59,101,187)(14,60,102,188)(15,61,103,189)(16,62,104,163)(17,63,105,164)(18,64,106,165)(19,65,107,166)(20,66,108,167)(21,67,82,168)(22,68,83,169)(23,69,84,170)(24,70,85,171)(25,71,86,172)(26,72,87,173)(27,73,88,174)(28,161,216,111)(29,162,190,112)(30,136,191,113)(31,137,192,114)(32,138,193,115)(33,139,194,116)(34,140,195,117)(35,141,196,118)(36,142,197,119)(37,143,198,120)(38,144,199,121)(39,145,200,122)(40,146,201,123)(41,147,202,124)(42,148,203,125)(43,149,204,126)(44,150,205,127)(45,151,206,128)(46,152,207,129)(47,153,208,130)(48,154,209,131)(49,155,210,132)(50,156,211,133)(51,157,212,134)(52,158,213,135)(53,159,214,109)(54,160,215,110), (28,216)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,189)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,180)(80,181)(81,182)>;
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), (1,154,89,131)(2,155,90,132)(3,156,91,133)(4,157,92,134)(5,158,93,135)(6,159,94,109)(7,160,95,110)(8,161,96,111)(9,162,97,112)(10,136,98,113)(11,137,99,114)(12,138,100,115)(13,139,101,116)(14,140,102,117)(15,141,103,118)(16,142,104,119)(17,143,105,120)(18,144,106,121)(19,145,107,122)(20,146,108,123)(21,147,82,124)(22,148,83,125)(23,149,84,126)(24,150,85,127)(25,151,86,128)(26,152,87,129)(27,153,88,130)(28,81,216,182)(29,55,190,183)(30,56,191,184)(31,57,192,185)(32,58,193,186)(33,59,194,187)(34,60,195,188)(35,61,196,189)(36,62,197,163)(37,63,198,164)(38,64,199,165)(39,65,200,166)(40,66,201,167)(41,67,202,168)(42,68,203,169)(43,69,204,170)(44,70,205,171)(45,71,206,172)(46,72,207,173)(47,73,208,174)(48,74,209,175)(49,75,210,176)(50,76,211,177)(51,77,212,178)(52,78,213,179)(53,79,214,180)(54,80,215,181), (1,74,89,175)(2,75,90,176)(3,76,91,177)(4,77,92,178)(5,78,93,179)(6,79,94,180)(7,80,95,181)(8,81,96,182)(9,55,97,183)(10,56,98,184)(11,57,99,185)(12,58,100,186)(13,59,101,187)(14,60,102,188)(15,61,103,189)(16,62,104,163)(17,63,105,164)(18,64,106,165)(19,65,107,166)(20,66,108,167)(21,67,82,168)(22,68,83,169)(23,69,84,170)(24,70,85,171)(25,71,86,172)(26,72,87,173)(27,73,88,174)(28,161,216,111)(29,162,190,112)(30,136,191,113)(31,137,192,114)(32,138,193,115)(33,139,194,116)(34,140,195,117)(35,141,196,118)(36,142,197,119)(37,143,198,120)(38,144,199,121)(39,145,200,122)(40,146,201,123)(41,147,202,124)(42,148,203,125)(43,149,204,126)(44,150,205,127)(45,151,206,128)(46,152,207,129)(47,153,208,130)(48,154,209,131)(49,155,210,132)(50,156,211,133)(51,157,212,134)(52,158,213,135)(53,159,214,109)(54,160,215,110), (28,216)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,183)(56,184)(57,185)(58,186)(59,187)(60,188)(61,189)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,169)(69,170)(70,171)(71,172)(72,173)(73,174)(74,175)(75,176)(76,177)(77,178)(78,179)(79,180)(80,181)(81,182) );
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)], [(1,154,89,131),(2,155,90,132),(3,156,91,133),(4,157,92,134),(5,158,93,135),(6,159,94,109),(7,160,95,110),(8,161,96,111),(9,162,97,112),(10,136,98,113),(11,137,99,114),(12,138,100,115),(13,139,101,116),(14,140,102,117),(15,141,103,118),(16,142,104,119),(17,143,105,120),(18,144,106,121),(19,145,107,122),(20,146,108,123),(21,147,82,124),(22,148,83,125),(23,149,84,126),(24,150,85,127),(25,151,86,128),(26,152,87,129),(27,153,88,130),(28,81,216,182),(29,55,190,183),(30,56,191,184),(31,57,192,185),(32,58,193,186),(33,59,194,187),(34,60,195,188),(35,61,196,189),(36,62,197,163),(37,63,198,164),(38,64,199,165),(39,65,200,166),(40,66,201,167),(41,67,202,168),(42,68,203,169),(43,69,204,170),(44,70,205,171),(45,71,206,172),(46,72,207,173),(47,73,208,174),(48,74,209,175),(49,75,210,176),(50,76,211,177),(51,77,212,178),(52,78,213,179),(53,79,214,180),(54,80,215,181)], [(1,74,89,175),(2,75,90,176),(3,76,91,177),(4,77,92,178),(5,78,93,179),(6,79,94,180),(7,80,95,181),(8,81,96,182),(9,55,97,183),(10,56,98,184),(11,57,99,185),(12,58,100,186),(13,59,101,187),(14,60,102,188),(15,61,103,189),(16,62,104,163),(17,63,105,164),(18,64,106,165),(19,65,107,166),(20,66,108,167),(21,67,82,168),(22,68,83,169),(23,69,84,170),(24,70,85,171),(25,71,86,172),(26,72,87,173),(27,73,88,174),(28,161,216,111),(29,162,190,112),(30,136,191,113),(31,137,192,114),(32,138,193,115),(33,139,194,116),(34,140,195,117),(35,141,196,118),(36,142,197,119),(37,143,198,120),(38,144,199,121),(39,145,200,122),(40,146,201,123),(41,147,202,124),(42,148,203,125),(43,149,204,126),(44,150,205,127),(45,151,206,128),(46,152,207,129),(47,153,208,130),(48,154,209,131),(49,155,210,132),(50,156,211,133),(51,157,212,134),(52,158,213,135),(53,159,214,109),(54,160,215,110)], [(28,216),(29,190),(30,191),(31,192),(32,193),(33,194),(34,195),(35,196),(36,197),(37,198),(38,199),(39,200),(40,201),(41,202),(42,203),(43,204),(44,205),(45,206),(46,207),(47,208),(48,209),(49,210),(50,211),(51,212),(52,213),(53,214),(54,215),(55,183),(56,184),(57,185),(58,186),(59,187),(60,188),(61,189),(62,163),(63,164),(64,165),(65,166),(66,167),(67,168),(68,169),(69,170),(70,171),(71,172),(72,173),(73,174),(74,175),(75,176),(76,177),(77,178),(78,179),(79,180),(80,181),(81,182)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 18A | ··· | 18F | 18G | ··· | 18X | 27A | ··· | 27R | 36A | ··· | 36L | 36M | ··· | 36AD | 54A | ··· | 54R | 54S | ··· | 54BT | 108A | ··· | 108AJ | 108AK | ··· | 108CL |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 36 | ··· | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 108 | ··· | 108 | 108 | ··· | 108 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | C27 | C54 | C54 | C54 | C4○D4 | C3×C4○D4 | C9×C4○D4 | C4○D4×C27 |
| kernel | C4○D4×C27 | C2×C108 | D4×C27 | Q8×C27 | C9×C4○D4 | C2×C36 | D4×C9 | Q8×C9 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C2×C4 | D4 | Q8 | C27 | C9 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 18 | 18 | 6 | 18 | 54 | 54 | 18 | 2 | 4 | 12 | 36 |
Matrix representation of C4○D4×C27 ►in GL2(𝔽109) generated by
| 15 | 0 |
| 0 | 15 |
| 33 | 0 |
| 0 | 33 |
| 108 | 65 |
| 5 | 1 |
| 1 | 0 |
| 104 | 108 |
G:=sub<GL(2,GF(109))| [15,0,0,15],[33,0,0,33],[108,5,65,1],[1,104,0,108] >;
C4○D4×C27 in GAP, Magma, Sage, TeX
C_4\circ D_4\times C_{27} % in TeX
G:=Group("C4oD4xC27"); // GroupNames label
G:=SmallGroup(432,56);
// by ID
G=gap.SmallGroup(432,56);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,142,192,166]);
// Polycyclic
G:=Group<a,b,c,d|a^27=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations