direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C43⋊C3, C86⋊C3, C43⋊2C6, SmallGroup(258,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C2×C43⋊C3 |
Generators and relations for C2×C43⋊C3
G = < a,b,c | a2=b43=c3=1, ab=ba, ac=ca, cbc-1=b6 >
Smallest permutation representation of C2×C43⋊C3
►On 86 points
Generators in S86
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 77)(35 78)(36 79)(37 80)(38 81)(39 82)(40 83)(41 84)(42 85)(43 86)
(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)
(2 37 7)(3 30 13)(4 23 19)(5 16 25)(6 9 31)(8 38 43)(10 24 12)(11 17 18)(14 39 36)(15 32 42)(20 40 29)(21 33 35)(22 26 41)(27 34 28)(45 80 50)(46 73 56)(47 66 62)(48 59 68)(49 52 74)(51 81 86)(53 67 55)(54 60 61)(57 82 79)(58 75 85)(63 83 72)(64 76 78)(65 69 84)(70 77 71)
G:=sub<Sym(86)| (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86), (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), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)(45,80,50)(46,73,56)(47,66,62)(48,59,68)(49,52,74)(51,81,86)(53,67,55)(54,60,61)(57,82,79)(58,75,85)(63,83,72)(64,76,78)(65,69,84)(70,77,71)>;
G:=Group( (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,77)(35,78)(36,79)(37,80)(38,81)(39,82)(40,83)(41,84)(42,85)(43,86), (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), (2,37,7)(3,30,13)(4,23,19)(5,16,25)(6,9,31)(8,38,43)(10,24,12)(11,17,18)(14,39,36)(15,32,42)(20,40,29)(21,33,35)(22,26,41)(27,34,28)(45,80,50)(46,73,56)(47,66,62)(48,59,68)(49,52,74)(51,81,86)(53,67,55)(54,60,61)(57,82,79)(58,75,85)(63,83,72)(64,76,78)(65,69,84)(70,77,71) );
G=PermutationGroup([[(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,77),(35,78),(36,79),(37,80),(38,81),(39,82),(40,83),(41,84),(42,85),(43,86)], [(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)], [(2,37,7),(3,30,13),(4,23,19),(5,16,25),(6,9,31),(8,38,43),(10,24,12),(11,17,18),(14,39,36),(15,32,42),(20,40,29),(21,33,35),(22,26,41),(27,34,28),(45,80,50),(46,73,56),(47,66,62),(48,59,68),(49,52,74),(51,81,86),(53,67,55),(54,60,61),(57,82,79),(58,75,85),(63,83,72),(64,76,78),(65,69,84),(70,77,71)]])
34 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 43A | ··· | 43N | 86A | ··· | 86N |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 43 | ··· | 43 | 86 | ··· | 86 |
| size | 1 | 1 | 43 | 43 | 43 | 43 | 3 | ··· | 3 | 3 | ··· | 3 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C43⋊C3 | C2×C43⋊C3 |
| kernel | C2×C43⋊C3 | C43⋊C3 | C86 | C43 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 14 | 14 |
Matrix representation of C2×C43⋊C3 ►in GL4(𝔽1033) generated by
| 1032 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 598 | 145 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 837 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 668 | 74 | 1009 |
| 0 | 701 | 963 | 958 |
G:=sub<GL(4,GF(1033))| [1032,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,598,1,0,0,145,0,1,0,1,0,0],[837,0,0,0,0,1,668,701,0,0,74,963,0,0,1009,958] >;
C2×C43⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{43}\rtimes C_3 % in TeX
G:=Group("C2xC43:C3"); // GroupNames label
G:=SmallGroup(258,2);
// by ID
G=gap.SmallGroup(258,2);
# by ID
G:=PCGroup([3,-2,-3,-43,977]);
// Polycyclic
G:=Group<a,b,c|a^2=b^43=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations
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