direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×D28, C28⋊4D6, D84⋊7C2, D14⋊1D6, C12⋊1D14, C84⋊2C22, Dic3⋊3D14, D6.10D14, D42⋊2C22, C42.13C23, C7⋊1(S3×D4), C4⋊1(S3×D7), C21⋊2(C2×D4), C3⋊1(C2×D28), (S3×C7)⋊1D4, (C4×S3)⋊3D7, (S3×C28)⋊3C2, (C3×D28)⋊3C2, C3⋊D28⋊3C2, (C6×D7)⋊1C22, C6.13(C22×D7), C14.13(C22×S3), (C7×Dic3)⋊4C22, (S3×C14).10C22, (C2×S3×D7)⋊2C2, C2.16(C2×S3×D7), SmallGroup(336,149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×D28
G = < a,b,c,d | a3=b2=c28=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 804 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C7, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, D7, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C28, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, S3×D4, D28, D28, C2×C28, C22×D7, C7×Dic3, C84, S3×D7, C6×D7, S3×C14, D42, C2×D28, C3⋊D28, C3×D28, S3×C28, D84, C2×S3×D7, S3×D28
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C22×S3, D14, S3×D4, D28, C22×D7, S3×D7, C2×D28, C2×S3×D7, S3×D28
►On 84 points
Generators in S84
(1 52 66)(2 53 67)(3 54 68)(4 55 69)(5 56 70)(6 29 71)(7 30 72)(8 31 73)(9 32 74)(10 33 75)(11 34 76)(12 35 77)(13 36 78)(14 37 79)(15 38 80)(16 39 81)(17 40 82)(18 41 83)(19 42 84)(20 43 57)(21 44 58)(22 45 59)(23 46 60)(24 47 61)(25 48 62)(26 49 63)(27 50 64)(28 51 65)
(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(54 56)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(82 84)
G:=sub<Sym(84)| (1,52,66)(2,53,67)(3,54,68)(4,55,69)(5,56,70)(6,29,71)(7,30,72)(8,31,73)(9,32,74)(10,33,75)(11,34,76)(12,35,77)(13,36,78)(14,37,79)(15,38,80)(16,39,81)(17,40,82)(18,41,83)(19,42,84)(20,43,57)(21,44,58)(22,45,59)(23,46,60)(24,47,61)(25,48,62)(26,49,63)(27,50,64)(28,51,65), (29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84)>;
G:=Group( (1,52,66)(2,53,67)(3,54,68)(4,55,69)(5,56,70)(6,29,71)(7,30,72)(8,31,73)(9,32,74)(10,33,75)(11,34,76)(12,35,77)(13,36,78)(14,37,79)(15,38,80)(16,39,81)(17,40,82)(18,41,83)(19,42,84)(20,43,57)(21,44,58)(22,45,59)(23,46,60)(24,47,61)(25,48,62)(26,49,63)(27,50,64)(28,51,65), (29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(54,56)(57,81)(58,80)(59,79)(60,78)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(82,84) );
G=PermutationGroup([[(1,52,66),(2,53,67),(3,54,68),(4,55,69),(5,56,70),(6,29,71),(7,30,72),(8,31,73),(9,32,74),(10,33,75),(11,34,76),(12,35,77),(13,36,78),(14,37,79),(15,38,80),(16,39,81),(17,40,82),(18,41,83),(19,42,84),(20,43,57),(21,44,58),(22,45,59),(23,46,60),(24,47,61),(25,48,62),(26,49,63),(27,50,64),(28,51,65)], [(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(54,56),(57,81),(58,80),(59,79),(60,78),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(82,84)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 12 | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | ··· | 28F | 28G | ··· | 28L | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 3 | 3 | 14 | 14 | 42 | 42 | 2 | 2 | 6 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D7 | D14 | D14 | D14 | D28 | S3×D4 | S3×D7 | C2×S3×D7 | S3×D28 |
| kernel | S3×D28 | C3⋊D28 | C3×D28 | S3×C28 | D84 | C2×S3×D7 | D28 | S3×C7 | C28 | D14 | C4×S3 | Dic3 | C12 | D6 | S3 | C7 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 3 | 3 | 3 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of S3×D28 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 335 | 141 |
| 0 | 0 | 43 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 294 | 336 |
| 24 | 299 | 0 | 0 |
| 38 | 319 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 109 | 143 | 0 | 0 |
| 228 | 228 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,335,43,0,0,141,1],[1,0,0,0,0,1,0,0,0,0,1,294,0,0,0,336],[24,38,0,0,299,319,0,0,0,0,1,0,0,0,0,1],[109,228,0,0,143,228,0,0,0,0,1,0,0,0,0,1] >;
S3×D28 in GAP, Magma, Sage, TeX
S_3\times D_{28} % in TeX
G:=Group("S3xD28"); // GroupNames label
G:=SmallGroup(336,149);
// by ID
G=gap.SmallGroup(336,149);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,50,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^28=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations