direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C27, C8⋊3C54, C4.C108, C216⋊7C2, C72.11C6, C108.4C4, C24.7C18, C12.5C36, C22.C108, C36.11C12, C108.22C22, (C2×C54).1C4, C4.5(C2×C54), (C2×C6).3C36, (C2×C4).2C54, C3.(C9×M4(2)), C9.(C3×M4(2)), C2.3(C2×C108), C36.51(C2×C6), (C2×C18).7C12, (C2×C108).8C2, (C2×C36).21C6, C6.12(C2×C36), (C2×C12).9C18, C54.12(C2×C4), (C9×M4(2)).C3, (C3×M4(2)).C9, C18.26(C2×C12), C12.28(C2×C18), SmallGroup(432,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C27
G = < a,b,c | a27=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×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)
(1 201 95 132 62 154 30 175)(2 202 96 133 63 155 31 176)(3 203 97 134 64 156 32 177)(4 204 98 135 65 157 33 178)(5 205 99 109 66 158 34 179)(6 206 100 110 67 159 35 180)(7 207 101 111 68 160 36 181)(8 208 102 112 69 161 37 182)(9 209 103 113 70 162 38 183)(10 210 104 114 71 136 39 184)(11 211 105 115 72 137 40 185)(12 212 106 116 73 138 41 186)(13 213 107 117 74 139 42 187)(14 214 108 118 75 140 43 188)(15 215 82 119 76 141 44 189)(16 216 83 120 77 142 45 163)(17 190 84 121 78 143 46 164)(18 191 85 122 79 144 47 165)(19 192 86 123 80 145 48 166)(20 193 87 124 81 146 49 167)(21 194 88 125 55 147 50 168)(22 195 89 126 56 148 51 169)(23 196 90 127 57 149 52 170)(24 197 91 128 58 150 53 171)(25 198 92 129 59 151 54 172)(26 199 93 130 60 152 28 173)(27 200 94 131 61 153 29 174)
(109 179)(110 180)(111 181)(112 182)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 163)(121 164)(122 165)(123 166)(124 167)(125 168)(126 169)(127 170)(128 171)(129 172)(130 173)(131 174)(132 175)(133 176)(134 177)(135 178)(136 210)(137 211)(138 212)(139 213)(140 214)(141 215)(142 216)(143 190)(144 191)(145 192)(146 193)(147 194)(148 195)(149 196)(150 197)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)
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), (1,201,95,132,62,154,30,175)(2,202,96,133,63,155,31,176)(3,203,97,134,64,156,32,177)(4,204,98,135,65,157,33,178)(5,205,99,109,66,158,34,179)(6,206,100,110,67,159,35,180)(7,207,101,111,68,160,36,181)(8,208,102,112,69,161,37,182)(9,209,103,113,70,162,38,183)(10,210,104,114,71,136,39,184)(11,211,105,115,72,137,40,185)(12,212,106,116,73,138,41,186)(13,213,107,117,74,139,42,187)(14,214,108,118,75,140,43,188)(15,215,82,119,76,141,44,189)(16,216,83,120,77,142,45,163)(17,190,84,121,78,143,46,164)(18,191,85,122,79,144,47,165)(19,192,86,123,80,145,48,166)(20,193,87,124,81,146,49,167)(21,194,88,125,55,147,50,168)(22,195,89,126,56,148,51,169)(23,196,90,127,57,149,52,170)(24,197,91,128,58,150,53,171)(25,198,92,129,59,151,54,172)(26,199,93,130,60,152,28,173)(27,200,94,131,61,153,29,174), (109,179)(110,180)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)(135,178)(136,210)(137,211)(138,212)(139,213)(140,214)(141,215)(142,216)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)>;
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), (1,201,95,132,62,154,30,175)(2,202,96,133,63,155,31,176)(3,203,97,134,64,156,32,177)(4,204,98,135,65,157,33,178)(5,205,99,109,66,158,34,179)(6,206,100,110,67,159,35,180)(7,207,101,111,68,160,36,181)(8,208,102,112,69,161,37,182)(9,209,103,113,70,162,38,183)(10,210,104,114,71,136,39,184)(11,211,105,115,72,137,40,185)(12,212,106,116,73,138,41,186)(13,213,107,117,74,139,42,187)(14,214,108,118,75,140,43,188)(15,215,82,119,76,141,44,189)(16,216,83,120,77,142,45,163)(17,190,84,121,78,143,46,164)(18,191,85,122,79,144,47,165)(19,192,86,123,80,145,48,166)(20,193,87,124,81,146,49,167)(21,194,88,125,55,147,50,168)(22,195,89,126,56,148,51,169)(23,196,90,127,57,149,52,170)(24,197,91,128,58,150,53,171)(25,198,92,129,59,151,54,172)(26,199,93,130,60,152,28,173)(27,200,94,131,61,153,29,174), (109,179)(110,180)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,163)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)(135,178)(136,210)(137,211)(138,212)(139,213)(140,214)(141,215)(142,216)(143,190)(144,191)(145,192)(146,193)(147,194)(148,195)(149,196)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209) );
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)], [(1,201,95,132,62,154,30,175),(2,202,96,133,63,155,31,176),(3,203,97,134,64,156,32,177),(4,204,98,135,65,157,33,178),(5,205,99,109,66,158,34,179),(6,206,100,110,67,159,35,180),(7,207,101,111,68,160,36,181),(8,208,102,112,69,161,37,182),(9,209,103,113,70,162,38,183),(10,210,104,114,71,136,39,184),(11,211,105,115,72,137,40,185),(12,212,106,116,73,138,41,186),(13,213,107,117,74,139,42,187),(14,214,108,118,75,140,43,188),(15,215,82,119,76,141,44,189),(16,216,83,120,77,142,45,163),(17,190,84,121,78,143,46,164),(18,191,85,122,79,144,47,165),(19,192,86,123,80,145,48,166),(20,193,87,124,81,146,49,167),(21,194,88,125,55,147,50,168),(22,195,89,126,56,148,51,169),(23,196,90,127,57,149,52,170),(24,197,91,128,58,150,53,171),(25,198,92,129,59,151,54,172),(26,199,93,130,60,152,28,173),(27,200,94,131,61,153,29,174)], [(109,179),(110,180),(111,181),(112,182),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,163),(121,164),(122,165),(123,166),(124,167),(125,168),(126,169),(127,170),(128,171),(129,172),(130,173),(131,174),(132,175),(133,176),(134,177),(135,178),(136,210),(137,211),(138,212),(139,213),(140,214),(141,215),(142,216),(143,190),(144,191),(145,192),(146,193),(147,194),(148,195),(149,196),(150,197),(151,198),(152,199),(153,200),(154,201),(155,202),(156,203),(157,204),(158,205),(159,206),(160,207),(161,208),(162,209)]])
270 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 18A | ··· | 18F | 18G | ··· | 18L | 24A | ··· | 24H | 27A | ··· | 27R | 36A | ··· | 36L | 36M | ··· | 36R | 54A | ··· | 54R | 54S | ··· | 54AJ | 72A | ··· | 72X | 108A | ··· | 108AJ | 108AK | ··· | 108BB | 216A | ··· | 216BT |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 36 | ··· | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
270 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C3 | C4 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C27 | C36 | C36 | C54 | C54 | C108 | C108 | M4(2) | C3×M4(2) | C9×M4(2) | M4(2)×C27 |
| kernel | M4(2)×C27 | C216 | C2×C108 | C9×M4(2) | C108 | C2×C54 | C72 | C2×C36 | C3×M4(2) | C36 | C2×C18 | C24 | C2×C12 | M4(2) | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C27 | C9 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 4 | 4 | 12 | 6 | 18 | 12 | 12 | 36 | 18 | 36 | 36 | 2 | 4 | 12 | 36 |
Matrix representation of M4(2)×C27 ►in GL2(𝔽433) generated by
| 269 | 0 |
| 0 | 269 |
| 254 | 431 |
| 343 | 179 |
| 1 | 0 |
| 254 | 432 |
G:=sub<GL(2,GF(433))| [269,0,0,269],[254,343,431,179],[1,254,0,432] >;
M4(2)×C27 in GAP, Magma, Sage, TeX
M_4(2)\times C_{27} % in TeX
G:=Group("M4(2)xC27"); // GroupNames label
G:=SmallGroup(432,24);
// by ID
G=gap.SmallGroup(432,24);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,1037,142,192,242]);
// Polycyclic
G:=Group<a,b,c|a^27=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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