direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C12×C7⋊C3, C84⋊C3, C28⋊C32, C21⋊4C12, C42.8C6, C7⋊2(C3×C12), C14.2(C3×C6), C2.(C6×C7⋊C3), C6.4(C2×C7⋊C3), (C6×C7⋊C3).4C2, (C2×C7⋊C3).2C6, SmallGroup(252,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C12×C7⋊C3 |
Generators and relations for C12×C7⋊C3
G = < a,b,c | a12=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C12×C7⋊C3
►On 84 points
Generators in S84
(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)
(1 83 24 32 43 66 59)(2 84 13 33 44 67 60)(3 73 14 34 45 68 49)(4 74 15 35 46 69 50)(5 75 16 36 47 70 51)(6 76 17 25 48 71 52)(7 77 18 26 37 72 53)(8 78 19 27 38 61 54)(9 79 20 28 39 62 55)(10 80 21 29 40 63 56)(11 81 22 30 41 64 57)(12 82 23 31 42 65 58)
(13 44 84)(14 45 73)(15 46 74)(16 47 75)(17 48 76)(18 37 77)(19 38 78)(20 39 79)(21 40 80)(22 41 81)(23 42 82)(24 43 83)(25 52 71)(26 53 72)(27 54 61)(28 55 62)(29 56 63)(30 57 64)(31 58 65)(32 59 66)(33 60 67)(34 49 68)(35 50 69)(36 51 70)
G:=sub<Sym(84)| (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), (1,83,24,32,43,66,59)(2,84,13,33,44,67,60)(3,73,14,34,45,68,49)(4,74,15,35,46,69,50)(5,75,16,36,47,70,51)(6,76,17,25,48,71,52)(7,77,18,26,37,72,53)(8,78,19,27,38,61,54)(9,79,20,28,39,62,55)(10,80,21,29,40,63,56)(11,81,22,30,41,64,57)(12,82,23,31,42,65,58), (13,44,84)(14,45,73)(15,46,74)(16,47,75)(17,48,76)(18,37,77)(19,38,78)(20,39,79)(21,40,80)(22,41,81)(23,42,82)(24,43,83)(25,52,71)(26,53,72)(27,54,61)(28,55,62)(29,56,63)(30,57,64)(31,58,65)(32,59,66)(33,60,67)(34,49,68)(35,50,69)(36,51,70)>;
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), (1,83,24,32,43,66,59)(2,84,13,33,44,67,60)(3,73,14,34,45,68,49)(4,74,15,35,46,69,50)(5,75,16,36,47,70,51)(6,76,17,25,48,71,52)(7,77,18,26,37,72,53)(8,78,19,27,38,61,54)(9,79,20,28,39,62,55)(10,80,21,29,40,63,56)(11,81,22,30,41,64,57)(12,82,23,31,42,65,58), (13,44,84)(14,45,73)(15,46,74)(16,47,75)(17,48,76)(18,37,77)(19,38,78)(20,39,79)(21,40,80)(22,41,81)(23,42,82)(24,43,83)(25,52,71)(26,53,72)(27,54,61)(28,55,62)(29,56,63)(30,57,64)(31,58,65)(32,59,66)(33,60,67)(34,49,68)(35,50,69)(36,51,70) );
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)], [(1,83,24,32,43,66,59),(2,84,13,33,44,67,60),(3,73,14,34,45,68,49),(4,74,15,35,46,69,50),(5,75,16,36,47,70,51),(6,76,17,25,48,71,52),(7,77,18,26,37,72,53),(8,78,19,27,38,61,54),(9,79,20,28,39,62,55),(10,80,21,29,40,63,56),(11,81,22,30,41,64,57),(12,82,23,31,42,65,58)], [(13,44,84),(14,45,73),(15,46,74),(16,47,75),(17,48,76),(18,37,77),(19,38,78),(20,39,79),(21,40,80),(22,41,81),(23,42,82),(24,43,83),(25,52,71),(26,53,72),(27,54,61),(28,55,62),(29,56,63),(30,57,64),(31,58,65),(32,59,66),(33,60,67),(34,49,68),(35,50,69),(36,51,70)]])
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 7A | 7B | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 14A | 14B | 21A | 21B | 21C | 21D | 28A | 28B | 28C | 28D | 42A | 42B | 42C | 42D | 84A | ··· | 84H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 7 | 7 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 14 | 14 | 21 | 21 | 21 | 21 | 28 | 28 | 28 | 28 | 42 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 3 | 3 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||||||
| image | C1 | C2 | C3 | C3 | C4 | C6 | C6 | C12 | C12 | C7⋊C3 | C2×C7⋊C3 | C3×C7⋊C3 | C4×C7⋊C3 | C6×C7⋊C3 | C12×C7⋊C3 |
| kernel | C12×C7⋊C3 | C6×C7⋊C3 | C4×C7⋊C3 | C84 | C3×C7⋊C3 | C2×C7⋊C3 | C42 | C7⋊C3 | C21 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 2 | 6 | 2 | 12 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C12×C7⋊C3 ►in GL3(𝔽337) generated by
| 265 | 0 | 0 |
| 0 | 265 | 0 |
| 0 | 0 | 265 |
| 0 | 0 | 1 |
| 1 | 0 | 213 |
| 0 | 1 | 212 |
| 1 | 0 | 212 |
| 0 | 0 | 336 |
| 0 | 1 | 336 |
G:=sub<GL(3,GF(337))| [265,0,0,0,265,0,0,0,265],[0,1,0,0,0,1,1,213,212],[1,0,0,0,0,1,212,336,336] >;
C12×C7⋊C3 in GAP, Magma, Sage, TeX
C_{12}\times C_7\rtimes C_3 % in TeX
G:=Group("C12xC7:C3"); // GroupNames label
G:=SmallGroup(252,19);
// by ID
G=gap.SmallGroup(252,19);
# by ID
G:=PCGroup([5,-2,-3,-3,-2,-7,90,909]);
// Polycyclic
G:=Group<a,b,c|a^12=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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