direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C12×3- 1+2, C36⋊C32, C12.2C33, C33.5C12, C9⋊2(C3×C12), (C3×C36)⋊4C3, (C3×C9)⋊11C12, C18.2(C3×C6), (C3×C18).19C6, C6.3(C32×C6), (C3×C12).8C32, (C32×C12).2C3, (C32×C6).13C6, C3.2(C32×C12), C2.(C6×3- 1+2), C32.11(C3×C12), C6.6(C2×3- 1+2), (C2×3- 1+2).6C6, (C6×3- 1+2).4C2, (C3×C6).11(C3×C6), SmallGroup(324,107)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×3- 1+2
G = < a,b,c | a12=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Subgroups: 150 in 114 conjugacy classes, 96 normal (15 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C9, C32, C32, C32, C12, C12, C12, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C36, C3×C12, C3×C12, C3×C12, C3×C18, C2×3- 1+2, C32×C6, C3×3- 1+2, C3×C36, C4×3- 1+2, C32×C12, C6×3- 1+2, C12×3- 1+2
Quotients: C1, C2, C3, C4, C6, C32, C12, C3×C6, 3- 1+2, C33, C3×C12, C2×3- 1+2, C32×C6, C3×3- 1+2, C4×3- 1+2, C32×C12, C6×3- 1+2, C12×3- 1+2
►On 108 points
Generators in S108
(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)
(1 45 21 93 57 73 25 61 98)(2 46 22 94 58 74 26 62 99)(3 47 23 95 59 75 27 63 100)(4 48 24 96 60 76 28 64 101)(5 37 13 85 49 77 29 65 102)(6 38 14 86 50 78 30 66 103)(7 39 15 87 51 79 31 67 104)(8 40 16 88 52 80 32 68 105)(9 41 17 89 53 81 33 69 106)(10 42 18 90 54 82 34 70 107)(11 43 19 91 55 83 35 71 108)(12 44 20 92 56 84 36 72 97)
(1 85 33)(2 86 34)(3 87 35)(4 88 36)(5 89 25)(6 90 26)(7 91 27)(8 92 28)(9 93 29)(10 94 30)(11 95 31)(12 96 32)(13 106 73)(14 107 74)(15 108 75)(16 97 76)(17 98 77)(18 99 78)(19 100 79)(20 101 80)(21 102 81)(22 103 82)(23 104 83)(24 105 84)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
G:=sub<Sym(108)| (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), (1,45,21,93,57,73,25,61,98)(2,46,22,94,58,74,26,62,99)(3,47,23,95,59,75,27,63,100)(4,48,24,96,60,76,28,64,101)(5,37,13,85,49,77,29,65,102)(6,38,14,86,50,78,30,66,103)(7,39,15,87,51,79,31,67,104)(8,40,16,88,52,80,32,68,105)(9,41,17,89,53,81,33,69,106)(10,42,18,90,54,82,34,70,107)(11,43,19,91,55,83,35,71,108)(12,44,20,92,56,84,36,72,97), (1,85,33)(2,86,34)(3,87,35)(4,88,36)(5,89,25)(6,90,26)(7,91,27)(8,92,28)(9,93,29)(10,94,30)(11,95,31)(12,96,32)(13,106,73)(14,107,74)(15,108,75)(16,97,76)(17,98,77)(18,99,78)(19,100,79)(20,101,80)(21,102,81)(22,103,82)(23,104,83)(24,105,84)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)>;
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), (1,45,21,93,57,73,25,61,98)(2,46,22,94,58,74,26,62,99)(3,47,23,95,59,75,27,63,100)(4,48,24,96,60,76,28,64,101)(5,37,13,85,49,77,29,65,102)(6,38,14,86,50,78,30,66,103)(7,39,15,87,51,79,31,67,104)(8,40,16,88,52,80,32,68,105)(9,41,17,89,53,81,33,69,106)(10,42,18,90,54,82,34,70,107)(11,43,19,91,55,83,35,71,108)(12,44,20,92,56,84,36,72,97), (1,85,33)(2,86,34)(3,87,35)(4,88,36)(5,89,25)(6,90,26)(7,91,27)(8,92,28)(9,93,29)(10,94,30)(11,95,31)(12,96,32)(13,106,73)(14,107,74)(15,108,75)(16,97,76)(17,98,77)(18,99,78)(19,100,79)(20,101,80)(21,102,81)(22,103,82)(23,104,83)(24,105,84)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72) );
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)], [(1,45,21,93,57,73,25,61,98),(2,46,22,94,58,74,26,62,99),(3,47,23,95,59,75,27,63,100),(4,48,24,96,60,76,28,64,101),(5,37,13,85,49,77,29,65,102),(6,38,14,86,50,78,30,66,103),(7,39,15,87,51,79,31,67,104),(8,40,16,88,52,80,32,68,105),(9,41,17,89,53,81,33,69,106),(10,42,18,90,54,82,34,70,107),(11,43,19,91,55,83,35,71,108),(12,44,20,92,56,84,36,72,97)], [(1,85,33),(2,86,34),(3,87,35),(4,88,36),(5,89,25),(6,90,26),(7,91,27),(8,92,28),(9,93,29),(10,94,30),(11,95,31),(12,96,32),(13,106,73),(14,107,74),(15,108,75),(16,97,76),(17,98,77),(18,99,78),(19,100,79),(20,101,80),(21,102,81),(22,103,82),(23,104,83),(24,105,84),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72)]])
132 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6N | 9A | ··· | 9R | 12A | ··· | 12P | 12Q | ··· | 12AB | 18A | ··· | 18R | 36A | ··· | 36AJ |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | 1 | 1 | ··· | 1 | 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 | 3 | 3 | 3 |
| type | + | + | |||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | 3- 1+2 | C2×3- 1+2 | C4×3- 1+2 |
| kernel | C12×3- 1+2 | C6×3- 1+2 | C3×C36 | C4×3- 1+2 | C32×C12 | C3×3- 1+2 | C3×C18 | C2×3- 1+2 | C32×C6 | C3×C9 | 3- 1+2 | C33 | C12 | C6 | C3 |
| # reps | 1 | 1 | 6 | 18 | 2 | 2 | 6 | 18 | 2 | 12 | 36 | 4 | 6 | 6 | 12 |
Matrix representation of C12×3- 1+2 ►in GL4(𝔽37) generated by
| 10 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 31 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 10 | 0 | 0 |
| 26 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(37))| [10,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[10,0,0,0,0,0,0,10,0,10,0,0,0,0,1,0],[26,0,0,0,0,26,0,0,0,0,10,0,0,0,0,1] >;
C12×3- 1+2 in GAP, Magma, Sage, TeX
C_{12}\times 3_-^{1+2} % in TeX
G:=Group("C12xES-(3,1)"); // GroupNames label
G:=SmallGroup(324,107);
// by ID
G=gap.SmallGroup(324,107);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,655,1034]);
// Polycyclic
G:=Group<a,b,c|a^12=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations