metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28⋊2D6, D28⋊4S3, D6⋊2D14, D21⋊1D4, D14⋊2D6, D12⋊4D7, C12⋊2D14, C84⋊4C22, C42.14C23, Dic21⋊6C22, D42.13C22, C3⋊2(D4×D7), C7⋊2(S3×D4), C4⋊3(S3×D7), C21⋊3(C2×D4), (C3×D28)⋊6C2, (C4×D21)⋊7C2, (C7×D12)⋊6C2, C21⋊D4⋊3C2, (C6×D7)⋊2C22, (S3×C14)⋊2C22, C6.14(C22×D7), C14.14(C22×S3), (C2×S3×D7)⋊3C2, C2.17(C2×S3×D7), SmallGroup(336,150)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28⋊D6
G = < a,b,c | a28=b6=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >
Subgroups: 756 in 108 conjugacy classes, 34 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, 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, Dic7, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, S3×D4, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, Dic21, C84, S3×D7, C6×D7, S3×C14, D42, D4×D7, C21⋊D4, C3×D28, C7×D12, C4×D21, C2×S3×D7, C28⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C22×S3, D14, S3×D4, C22×D7, S3×D7, D4×D7, C2×S3×D7, C28⋊D6
►On 84 points
(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 46 70 22 53 63)(2 45 71 21 54 62)(3 44 72 20 55 61)(4 43 73 19 56 60)(5 42 74 18 29 59)(6 41 75 17 30 58)(7 40 76 16 31 57)(8 39 77 15 32 84)(9 38 78 14 33 83)(10 37 79 13 34 82)(11 36 80 12 35 81)(23 52 64 28 47 69)(24 51 65 27 48 68)(25 50 66 26 49 67)
(1 84)(2 69)(3 82)(4 67)(5 80)(6 65)(7 78)(8 63)(9 76)(10 61)(11 74)(12 59)(13 72)(14 57)(15 70)(16 83)(17 68)(18 81)(19 66)(20 79)(21 64)(22 77)(23 62)(24 75)(25 60)(26 73)(27 58)(28 71)(29 35)(30 48)(31 33)(32 46)(34 44)(36 42)(37 55)(38 40)(39 53)(41 51)(43 49)(45 47)(50 56)(52 54)
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), (1,46,70,22,53,63)(2,45,71,21,54,62)(3,44,72,20,55,61)(4,43,73,19,56,60)(5,42,74,18,29,59)(6,41,75,17,30,58)(7,40,76,16,31,57)(8,39,77,15,32,84)(9,38,78,14,33,83)(10,37,79,13,34,82)(11,36,80,12,35,81)(23,52,64,28,47,69)(24,51,65,27,48,68)(25,50,66,26,49,67), (1,84)(2,69)(3,82)(4,67)(5,80)(6,65)(7,78)(8,63)(9,76)(10,61)(11,74)(12,59)(13,72)(14,57)(15,70)(16,83)(17,68)(18,81)(19,66)(20,79)(21,64)(22,77)(23,62)(24,75)(25,60)(26,73)(27,58)(28,71)(29,35)(30,48)(31,33)(32,46)(34,44)(36,42)(37,55)(38,40)(39,53)(41,51)(43,49)(45,47)(50,56)(52,54)>;
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), (1,46,70,22,53,63)(2,45,71,21,54,62)(3,44,72,20,55,61)(4,43,73,19,56,60)(5,42,74,18,29,59)(6,41,75,17,30,58)(7,40,76,16,31,57)(8,39,77,15,32,84)(9,38,78,14,33,83)(10,37,79,13,34,82)(11,36,80,12,35,81)(23,52,64,28,47,69)(24,51,65,27,48,68)(25,50,66,26,49,67), (1,84)(2,69)(3,82)(4,67)(5,80)(6,65)(7,78)(8,63)(9,76)(10,61)(11,74)(12,59)(13,72)(14,57)(15,70)(16,83)(17,68)(18,81)(19,66)(20,79)(21,64)(22,77)(23,62)(24,75)(25,60)(26,73)(27,58)(28,71)(29,35)(30,48)(31,33)(32,46)(34,44)(36,42)(37,55)(38,40)(39,53)(41,51)(43,49)(45,47)(50,56)(52,54) );
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)], [(1,46,70,22,53,63),(2,45,71,21,54,62),(3,44,72,20,55,61),(4,43,73,19,56,60),(5,42,74,18,29,59),(6,41,75,17,30,58),(7,40,76,16,31,57),(8,39,77,15,32,84),(9,38,78,14,33,83),(10,37,79,13,34,82),(11,36,80,12,35,81),(23,52,64,28,47,69),(24,51,65,27,48,68),(25,50,66,26,49,67)], [(1,84),(2,69),(3,82),(4,67),(5,80),(6,65),(7,78),(8,63),(9,76),(10,61),(11,74),(12,59),(13,72),(14,57),(15,70),(16,83),(17,68),(18,81),(19,66),(20,79),(21,64),(22,77),(23,62),(24,75),(25,60),(26,73),(27,58),(28,71),(29,35),(30,48),(31,33),(32,46),(34,44),(36,42),(37,55),(38,40),(39,53),(41,51),(43,49),(45,47),(50,56),(52,54)]])
42 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 | 28B | 28C | 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 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 6 | 14 | 14 | 21 | 21 | 2 | 2 | 42 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D7 | D14 | D14 | S3×D4 | S3×D7 | D4×D7 | C2×S3×D7 | C28⋊D6 |
| kernel | C28⋊D6 | C21⋊D4 | C3×D28 | C7×D12 | C4×D21 | C2×S3×D7 | D28 | D21 | C28 | D14 | D12 | C12 | D6 | C7 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 3 | 6 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of C28⋊D6 ►in GL6(𝔽337)
| 336 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 110 | 0 | 0 |
| 0 | 0 | 227 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 335 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 336 | 7 | 0 | 0 | 0 | 0 |
| 144 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 304 | 227 | 0 | 0 |
| 0 | 0 | 304 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 335 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 2 | 330 | 0 | 0 | 0 | 0 |
| 193 | 335 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 110 | 0 | 0 |
| 0 | 0 | 33 | 304 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,33,227,0,0,0,0,110,1,0,0,0,0,0,0,336,1,0,0,0,0,335,1],[336,144,0,0,0,0,7,2,0,0,0,0,0,0,304,304,0,0,0,0,227,33,0,0,0,0,0,0,336,0,0,0,0,0,335,1],[2,193,0,0,0,0,330,335,0,0,0,0,0,0,33,33,0,0,0,0,110,304,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C28⋊D6 in GAP, Magma, Sage, TeX
C_{28}\rtimes D_6 % in TeX
G:=Group("C28:D6"); // GroupNames label
G:=SmallGroup(336,150);
// by ID
G=gap.SmallGroup(336,150);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c|a^28=b^6=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations