direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D28, C21⋊5D4, C28⋊5C6, C84⋊3C2, C12⋊3D7, D14⋊4C6, C6.15D14, C42.15C22, C4⋊(C3×D7), C7⋊4(C3×D4), (C6×D7)⋊4C2, C2.4(C6×D7), C14.11(C2×C6), SmallGroup(168,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D28
G = < a,b,c | a3=b28=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D28
►On 84 points
Generators in S84
(1 40 71)(2 41 72)(3 42 73)(4 43 74)(5 44 75)(6 45 76)(7 46 77)(8 47 78)(9 48 79)(10 49 80)(11 50 81)(12 51 82)(13 52 83)(14 53 84)(15 54 57)(16 55 58)(17 56 59)(18 29 60)(19 30 61)(20 31 62)(21 32 63)(22 33 64)(23 34 65)(24 35 66)(25 36 67)(26 37 68)(27 38 69)(28 39 70)
(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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(51 56)(52 55)(53 54)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)
G:=sub<Sym(84)| (1,40,71)(2,41,72)(3,42,73)(4,43,74)(5,44,75)(6,45,76)(7,46,77)(8,47,78)(9,48,79)(10,49,80)(11,50,81)(12,51,82)(13,52,83)(14,53,84)(15,54,57)(16,55,58)(17,56,59)(18,29,60)(19,30,61)(20,31,62)(21,32,63)(22,33,64)(23,34,65)(24,35,66)(25,36,67)(26,37,68)(27,38,69)(28,39,70), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,56)(52,55)(53,54)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)>;
G:=Group( (1,40,71)(2,41,72)(3,42,73)(4,43,74)(5,44,75)(6,45,76)(7,46,77)(8,47,78)(9,48,79)(10,49,80)(11,50,81)(12,51,82)(13,52,83)(14,53,84)(15,54,57)(16,55,58)(17,56,59)(18,29,60)(19,30,61)(20,31,62)(21,32,63)(22,33,64)(23,34,65)(24,35,66)(25,36,67)(26,37,68)(27,38,69)(28,39,70), (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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(51,56)(52,55)(53,54)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71) );
G=PermutationGroup([[(1,40,71),(2,41,72),(3,42,73),(4,43,74),(5,44,75),(6,45,76),(7,46,77),(8,47,78),(9,48,79),(10,49,80),(11,50,81),(12,51,82),(13,52,83),(14,53,84),(15,54,57),(16,55,58),(17,56,59),(18,29,60),(19,30,61),(20,31,62),(21,32,63),(22,33,64),(23,34,65),(24,35,66),(25,36,67),(26,37,68),(27,38,69),(28,39,70)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(51,56),(52,55),(53,54),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71)]])
C3×D28 is a maximal subgroup of
C21⋊D8 C3⋊D56 C28.D6 C6.D28 D28⋊5S3 D28⋊S3 C28⋊D6 C3×D4×D7
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 14 | 14 | 1 | 1 | 2 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | D7 | C3×D4 | D14 | C3×D7 | D28 | C6×D7 | C3×D28 |
| kernel | C3×D28 | C84 | C6×D7 | D28 | C28 | D14 | C21 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C3×D28 ►in GL2(𝔽337) generated by
| 128 | 0 |
| 0 | 128 |
| 313 | 275 |
| 124 | 222 |
| 305 | 281 |
| 265 | 32 |
G:=sub<GL(2,GF(337))| [128,0,0,128],[313,124,275,222],[305,265,281,32] >;
C3×D28 in GAP, Magma, Sage, TeX
C_3\times D_{28} % in TeX
G:=Group("C3xD28"); // GroupNames label
G:=SmallGroup(168,26);
// by ID
G=gap.SmallGroup(168,26);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,141,66,3604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^28=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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