metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D21⋊2D4, D14⋊4D6, D6⋊4D14, Dic7⋊2D6, Dic3⋊2D14, D42⋊10C22, C42.27C23, C7⋊3(S3×D4), C3⋊3(D4×D7), C21⋊9(C2×D4), (C2×C6)⋊2D14, (C2×C14)⋊5D6, C7⋊D4⋊2S3, C3⋊D4⋊2D7, D21⋊C4⋊5C2, C7⋊D12⋊6C2, C3⋊D28⋊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), (C7×Dic3)⋊2C22, (C3×Dic7)⋊2C22, (C2×S3×D7)⋊6C2, C2.27(C2×S3×D7), (C7×C3⋊D4)⋊4C2, (C3×C7⋊D4)⋊4C2, SmallGroup(336,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D21⋊D4
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 [×6], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6, C6 [×2], C7, C2×C4, D4 [×4], C23 [×2], Dic3, C12, D6, D6 [×6], C2×C6, C2×C6, D7 [×4], C14, C14 [×2], C2×D4, C21, C4×S3, D12, C3⋊D4, C3⋊D4, C3×D4, C22×S3 [×2], Dic7, C28, D14, D14 [×6], C2×C14, C2×C14, S3×C7, C3×D7, D21 [×2], D21, C42, C42, S3×D4, C4×D7, D28, C7⋊D4, C7⋊D4, C7×D4, C22×D7 [×2], C7×Dic3, C3×Dic7, S3×D7 [×2], C6×D7, S3×C14, D42 [×2], D42 [×2], C2×C42, D4×D7, D21⋊C4, C3⋊D28, C7⋊D12, C3×C7⋊D4, C7×C3⋊D4, C2×S3×D7, C22×D21, D21⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], D7, C2×D4, C22×S3, D14 [×3], S3×D4, C22×D7, S3×D7, D4×D7, C2×S3×D7, D21⋊D4
►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 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(21 42)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)
(1 54 42 82)(2 62 22 69)(3 49 23 77)(4 57 24 64)(5 44 25 72)(6 52 26 80)(7 60 27 67)(8 47 28 75)(9 55 29 83)(10 63 30 70)(11 50 31 78)(12 58 32 65)(13 45 33 73)(14 53 34 81)(15 61 35 68)(16 48 36 76)(17 56 37 84)(18 43 38 71)(19 51 39 79)(20 59 40 66)(21 46 41 74)
(2 9)(3 17)(5 12)(6 20)(8 15)(11 18)(14 21)(22 29)(23 37)(25 32)(26 40)(28 35)(31 38)(34 41)(43 78)(44 65)(45 73)(46 81)(47 68)(48 76)(49 84)(50 71)(51 79)(52 66)(53 74)(54 82)(55 69)(56 77)(57 64)(58 72)(59 80)(60 67)(61 75)(62 83)(63 70)
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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,42)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72), (1,54,42,82)(2,62,22,69)(3,49,23,77)(4,57,24,64)(5,44,25,72)(6,52,26,80)(7,60,27,67)(8,47,28,75)(9,55,29,83)(10,63,30,70)(11,50,31,78)(12,58,32,65)(13,45,33,73)(14,53,34,81)(15,61,35,68)(16,48,36,76)(17,56,37,84)(18,43,38,71)(19,51,39,79)(20,59,40,66)(21,46,41,74), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,29)(23,37)(25,32)(26,40)(28,35)(31,38)(34,41)(43,78)(44,65)(45,73)(46,81)(47,68)(48,76)(49,84)(50,71)(51,79)(52,66)(53,74)(54,82)(55,69)(56,77)(57,64)(58,72)(59,80)(60,67)(61,75)(62,83)(63,70)>;
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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(21,42)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72), (1,54,42,82)(2,62,22,69)(3,49,23,77)(4,57,24,64)(5,44,25,72)(6,52,26,80)(7,60,27,67)(8,47,28,75)(9,55,29,83)(10,63,30,70)(11,50,31,78)(12,58,32,65)(13,45,33,73)(14,53,34,81)(15,61,35,68)(16,48,36,76)(17,56,37,84)(18,43,38,71)(19,51,39,79)(20,59,40,66)(21,46,41,74), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,29)(23,37)(25,32)(26,40)(28,35)(31,38)(34,41)(43,78)(44,65)(45,73)(46,81)(47,68)(48,76)(49,84)(50,71)(51,79)(52,66)(53,74)(54,82)(55,69)(56,77)(57,64)(58,72)(59,80)(60,67)(61,75)(62,83)(63,70) );
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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(21,42),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72)], [(1,54,42,82),(2,62,22,69),(3,49,23,77),(4,57,24,64),(5,44,25,72),(6,52,26,80),(7,60,27,67),(8,47,28,75),(9,55,29,83),(10,63,30,70),(11,50,31,78),(12,58,32,65),(13,45,33,73),(14,53,34,81),(15,61,35,68),(16,48,36,76),(17,56,37,84),(18,43,38,71),(19,51,39,79),(20,59,40,66),(21,46,41,74)], [(2,9),(3,17),(5,12),(6,20),(8,15),(11,18),(14,21),(22,29),(23,37),(25,32),(26,40),(28,35),(31,38),(34,41),(43,78),(44,65),(45,73),(46,81),(47,68),(48,76),(49,84),(50,71),(51,79),(52,66),(53,74),(54,82),(55,69),(56,77),(57,64),(58,72),(59,80),(60,67),(61,75),(62,83),(63,70)])
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 | D21⋊D4 |
| kernel | D21⋊D4 | 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 D21⋊D4 ►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] >;
D21⋊D4 in GAP, Magma, Sage, TeX
D_{21}\rtimes D_4 % in TeX
G:=Group("D21:D4"); // 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