metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊4D14, D14⋊4D6, D21⋊2D4, Dic7⋊2D6, Dic3⋊2D14, D42⋊10C22, C42.27C23, C7⋊3(S3×D4), C3⋊3(D4×D7), C21⋊9(C2×D4), (C2×C14)⋊5D6, (C2×C6)⋊2D14, C3⋊D4⋊2D7, C7⋊D4⋊2S3, D21⋊C4⋊5C2, C3⋊D28⋊6C2, C7⋊D12⋊6C2, C22⋊4(S3×D7), (C2×C42)⋊4C22, (C6×D7)⋊4C22, (S3×C14)⋊4C22, (C22×D21)⋊6C2, C6.27(C22×D7), C14.27(C22×S3), (C3×Dic7)⋊2C22, (C7×Dic3)⋊2C22, (C2×S3×D7)⋊6C2, C2.27(C2×S3×D7), (C3×C7⋊D4)⋊4C2, (C7×C3⋊D4)⋊4C2, SmallGroup(336,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊D14
G = < a,b,c,d | a21=b2=c4=d2=1, bab=a-1, cac-1=dad=a8, cbc-1=dbd=a7b, dcd=c-1 >
Subgroups: 780 in 108 conjugacy classes, 34 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, D7, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3, Dic7, C28, D14, D14, C2×C14, C2×C14, S3×C7, C3×D7, D21, D21, C42, C42, S3×D4, C4×D7, D28, C7⋊D4, C7⋊D4, C7×D4, C22×D7, C7×Dic3, C3×Dic7, S3×D7, C6×D7, S3×C14, D42, D42, C2×C42, D4×D7, D21⋊C4, C3⋊D28, C7⋊D12, C3×C7⋊D4, C7×C3⋊D4, C2×S3×D7, C22×D21, D6⋊D14
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, D6⋊D14
►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 22)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 36)(9 35)(10 34)(11 33)(12 32)(13 31)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(21 23)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 84)(60 83)(61 82)(62 81)(63 80)
(1 52 23 71)(2 60 24 79)(3 47 25 66)(4 55 26 74)(5 63 27 82)(6 50 28 69)(7 58 29 77)(8 45 30 64)(9 53 31 72)(10 61 32 80)(11 48 33 67)(12 56 34 75)(13 43 35 83)(14 51 36 70)(15 59 37 78)(16 46 38 65)(17 54 39 73)(18 62 40 81)(19 49 41 68)(20 57 42 76)(21 44 22 84)
(2 9)(3 17)(5 12)(6 20)(8 15)(11 18)(14 21)(22 36)(24 31)(25 39)(27 34)(28 42)(30 37)(33 40)(43 83)(44 70)(45 78)(46 65)(47 73)(48 81)(49 68)(50 76)(51 84)(52 71)(53 79)(54 66)(55 74)(56 82)(57 69)(58 77)(59 64)(60 72)(61 80)(62 67)(63 75)
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), (1,22)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,84)(60,83)(61,82)(62,81)(63,80), (1,52,23,71)(2,60,24,79)(3,47,25,66)(4,55,26,74)(5,63,27,82)(6,50,28,69)(7,58,29,77)(8,45,30,64)(9,53,31,72)(10,61,32,80)(11,48,33,67)(12,56,34,75)(13,43,35,83)(14,51,36,70)(15,59,37,78)(16,46,38,65)(17,54,39,73)(18,62,40,81)(19,49,41,68)(20,57,42,76)(21,44,22,84), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,36)(24,31)(25,39)(27,34)(28,42)(30,37)(33,40)(43,83)(44,70)(45,78)(46,65)(47,73)(48,81)(49,68)(50,76)(51,84)(52,71)(53,79)(54,66)(55,74)(56,82)(57,69)(58,77)(59,64)(60,72)(61,80)(62,67)(63,75)>;
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), (1,22)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,36)(9,35)(10,34)(11,33)(12,32)(13,31)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(21,23)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,84)(60,83)(61,82)(62,81)(63,80), (1,52,23,71)(2,60,24,79)(3,47,25,66)(4,55,26,74)(5,63,27,82)(6,50,28,69)(7,58,29,77)(8,45,30,64)(9,53,31,72)(10,61,32,80)(11,48,33,67)(12,56,34,75)(13,43,35,83)(14,51,36,70)(15,59,37,78)(16,46,38,65)(17,54,39,73)(18,62,40,81)(19,49,41,68)(20,57,42,76)(21,44,22,84), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,36)(24,31)(25,39)(27,34)(28,42)(30,37)(33,40)(43,83)(44,70)(45,78)(46,65)(47,73)(48,81)(49,68)(50,76)(51,84)(52,71)(53,79)(54,66)(55,74)(56,82)(57,69)(58,77)(59,64)(60,72)(61,80)(62,67)(63,75) );
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)], [(1,22),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,36),(9,35),(10,34),(11,33),(12,32),(13,31),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(21,23),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,84),(60,83),(61,82),(62,81),(63,80)], [(1,52,23,71),(2,60,24,79),(3,47,25,66),(4,55,26,74),(5,63,27,82),(6,50,28,69),(7,58,29,77),(8,45,30,64),(9,53,31,72),(10,61,32,80),(11,48,33,67),(12,56,34,75),(13,43,35,83),(14,51,36,70),(15,59,37,78),(16,46,38,65),(17,54,39,73),(18,62,40,81),(19,49,41,68),(20,57,42,76),(21,44,22,84)], [(2,9),(3,17),(5,12),(6,20),(8,15),(11,18),(14,21),(22,36),(24,31),(25,39),(27,34),(28,42),(30,37),(33,40),(43,83),(44,70),(45,78),(46,65),(47,73),(48,81),(49,68),(50,76),(51,84),(52,71),(53,79),(54,66),(55,74),(56,82),(57,69),(58,77),(59,64),(60,72),(61,80),(62,67),(63,75)]])
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 | 14E | 14F | 14G | 14H | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 6 | 14 | 21 | 21 | 42 | 2 | 6 | 14 | 2 | 4 | 28 | 2 | 2 | 2 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | 4 | 4 | 12 | 12 | 12 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D7 | D14 | D14 | D14 | S3×D4 | S3×D7 | D4×D7 | C2×S3×D7 | D6⋊D14 |
| kernel | D6⋊D14 | D21⋊C4 | C3⋊D28 | C7⋊D12 | C3×C7⋊D4 | C7×C3⋊D4 | C2×S3×D7 | C22×D21 | C7⋊D4 | D21 | Dic7 | D14 | C2×C14 | C3⋊D4 | Dic3 | D6 | C2×C6 | C7 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 3 | 3 | 3 | 6 |
Matrix representation of D6⋊D14 ►in GL6(𝔽337)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 336 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 193 | 194 | 0 | 0 |
| 0 | 0 | 178 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 144 | 1 | 0 | 0 |
| 0 | 0 | 159 | 193 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 336 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 155 | 290 |
| 0 | 0 | 0 | 0 | 124 | 182 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 336 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 | 0 | 0 |
| 0 | 0 | 0 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 150 | 336 |
G:=sub<GL(6,GF(337))| [0,336,0,0,0,0,1,336,0,0,0,0,0,0,193,178,0,0,0,0,194,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,144,159,0,0,0,0,1,193,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[336,1,0,0,0,0,0,1,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,155,124,0,0,0,0,290,182],[1,336,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,336,0,0,0,0,0,0,1,150,0,0,0,0,0,336] >;
D6⋊D14 in GAP, Magma, Sage, TeX
D_6\rtimes D_{14} % in TeX
G:=Group("D6:D14"); // GroupNames label
G:=SmallGroup(336,163);
// by ID
G=gap.SmallGroup(336,163);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,116,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^8,c*b*c^-1=d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations