direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C4×C27⋊C3, C108⋊C3, C36.C9, C9.C36, C27⋊2C12, C54.2C6, C32.C36, C18.3C18, C36.2C32, (C3×C12).C9, (C3×C36).3C3, C3.3(C3×C36), C12.3(C3×C9), C9.1(C3×C12), (C3×C9).4C12, C18.5(C3×C6), C6.4(C3×C18), (C3×C6).3C18, (C3×C18).17C6, C2.(C2×C27⋊C3), (C2×C27⋊C3).2C2, SmallGroup(324,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C27⋊C3
G = < a,b,c | a4=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C4×C27⋊C3
►On 108 points
Generators in S108
(1 88 53 73)(2 89 54 74)(3 90 28 75)(4 91 29 76)(5 92 30 77)(6 93 31 78)(7 94 32 79)(8 95 33 80)(9 96 34 81)(10 97 35 55)(11 98 36 56)(12 99 37 57)(13 100 38 58)(14 101 39 59)(15 102 40 60)(16 103 41 61)(17 104 42 62)(18 105 43 63)(19 106 44 64)(20 107 45 65)(21 108 46 66)(22 82 47 67)(23 83 48 68)(24 84 49 69)(25 85 50 70)(26 86 51 71)(27 87 52 72)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(28 37 46)(30 48 39)(31 40 49)(33 51 42)(34 43 52)(36 54 45)(56 74 65)(57 66 75)(59 77 68)(60 69 78)(62 80 71)(63 72 81)(83 101 92)(84 93 102)(86 104 95)(87 96 105)(89 107 98)(90 99 108)
G:=sub<Sym(108)| (1,88,53,73)(2,89,54,74)(3,90,28,75)(4,91,29,76)(5,92,30,77)(6,93,31,78)(7,94,32,79)(8,95,33,80)(9,96,34,81)(10,97,35,55)(11,98,36,56)(12,99,37,57)(13,100,38,58)(14,101,39,59)(15,102,40,60)(16,103,41,61)(17,104,42,62)(18,105,43,63)(19,106,44,64)(20,107,45,65)(21,108,46,66)(22,82,47,67)(23,83,48,68)(24,84,49,69)(25,85,50,70)(26,86,51,71)(27,87,52,72), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(28,37,46)(30,48,39)(31,40,49)(33,51,42)(34,43,52)(36,54,45)(56,74,65)(57,66,75)(59,77,68)(60,69,78)(62,80,71)(63,72,81)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108)>;
G:=Group( (1,88,53,73)(2,89,54,74)(3,90,28,75)(4,91,29,76)(5,92,30,77)(6,93,31,78)(7,94,32,79)(8,95,33,80)(9,96,34,81)(10,97,35,55)(11,98,36,56)(12,99,37,57)(13,100,38,58)(14,101,39,59)(15,102,40,60)(16,103,41,61)(17,104,42,62)(18,105,43,63)(19,106,44,64)(20,107,45,65)(21,108,46,66)(22,82,47,67)(23,83,48,68)(24,84,49,69)(25,85,50,70)(26,86,51,71)(27,87,52,72), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(28,37,46)(30,48,39)(31,40,49)(33,51,42)(34,43,52)(36,54,45)(56,74,65)(57,66,75)(59,77,68)(60,69,78)(62,80,71)(63,72,81)(83,101,92)(84,93,102)(86,104,95)(87,96,105)(89,107,98)(90,99,108) );
G=PermutationGroup([[(1,88,53,73),(2,89,54,74),(3,90,28,75),(4,91,29,76),(5,92,30,77),(6,93,31,78),(7,94,32,79),(8,95,33,80),(9,96,34,81),(10,97,35,55),(11,98,36,56),(12,99,37,57),(13,100,38,58),(14,101,39,59),(15,102,40,60),(16,103,41,61),(17,104,42,62),(18,105,43,63),(19,106,44,64),(20,107,45,65),(21,108,46,66),(22,82,47,67),(23,83,48,68),(24,84,49,69),(25,85,50,70),(26,86,51,71),(27,87,52,72)], [(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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(28,37,46),(30,48,39),(31,40,49),(33,51,42),(34,43,52),(36,54,45),(56,74,65),(57,66,75),(59,77,68),(60,69,78),(62,80,71),(63,72,81),(83,101,92),(84,93,102),(86,104,95),(87,96,105),(89,107,98),(90,99,108)]])
132 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18F | 18G | 18H | 18I | 18J | 27A | ··· | 27R | 36A | ··· | 36L | 36M | ··· | 36T | 54A | ··· | 54R | 108A | ··· | 108AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | 18 | 18 | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C9 | C9 | C12 | C12 | C18 | C18 | C36 | C36 | C27⋊C3 | C2×C27⋊C3 | C4×C27⋊C3 |
| kernel | C4×C27⋊C3 | C2×C27⋊C3 | C108 | C3×C36 | C27⋊C3 | C54 | C3×C18 | C36 | C3×C12 | C27 | C3×C9 | C18 | C3×C6 | C9 | C32 | C4 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 6 | 12 | 4 | 12 | 6 | 24 | 12 | 6 | 6 | 12 |
Matrix representation of C4×C27⋊C3 ►in GL4(𝔽109) generated by
| 76 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 45 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 45 |
| 0 | 75 | 0 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 63 |
G:=sub<GL(4,GF(109))| [76,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[45,0,0,0,0,0,0,75,0,1,0,0,0,0,45,0],[63,0,0,0,0,1,0,0,0,0,45,0,0,0,0,63] >;
C4×C27⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_{27}\rtimes C_3 % in TeX
G:=Group("C4xC27:C3"); // GroupNames label
G:=SmallGroup(324,30);
// by ID
G=gap.SmallGroup(324,30);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,223,1034,118]);
// Polycyclic
G:=Group<a,b,c|a^4=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
Export