metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C21⋊7D4, D42⋊2C2, C2.5D42, C14.12D6, C6.12D14, C22⋊2D21, Dic21⋊1C2, C42.12C22, (C2×C6)⋊2D7, (C2×C14)⋊4S3, (C2×C42)⋊2C2, C3⋊3(C7⋊D4), C7⋊3(C3⋊D4), SmallGroup(168,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C21⋊7D4
G = < a,b,c | a21=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C21⋊7D4
►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 46 37 74)(2 45 38 73)(3 44 39 72)(4 43 40 71)(5 63 41 70)(6 62 42 69)(7 61 22 68)(8 60 23 67)(9 59 24 66)(10 58 25 65)(11 57 26 64)(12 56 27 84)(13 55 28 83)(14 54 29 82)(15 53 30 81)(16 52 31 80)(17 51 32 79)(18 50 33 78)(19 49 34 77)(20 48 35 76)(21 47 36 75)
(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(11 12)(22 31)(23 30)(24 29)(25 28)(26 27)(32 42)(33 41)(34 40)(35 39)(36 38)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)
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), (1,46,37,74)(2,45,38,73)(3,44,39,72)(4,43,40,71)(5,63,41,70)(6,62,42,69)(7,61,22,68)(8,60,23,67)(9,59,24,66)(10,58,25,65)(11,57,26,64)(12,56,27,84)(13,55,28,83)(14,54,29,82)(15,53,30,81)(16,52,31,80)(17,51,32,79)(18,50,33,78)(19,49,34,77)(20,48,35,76)(21,47,36,75), (2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(22,31)(23,30)(24,29)(25,28)(26,27)(32,42)(33,41)(34,40)(35,39)(36,38)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)>;
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), (1,46,37,74)(2,45,38,73)(3,44,39,72)(4,43,40,71)(5,63,41,70)(6,62,42,69)(7,61,22,68)(8,60,23,67)(9,59,24,66)(10,58,25,65)(11,57,26,64)(12,56,27,84)(13,55,28,83)(14,54,29,82)(15,53,30,81)(16,52,31,80)(17,51,32,79)(18,50,33,78)(19,49,34,77)(20,48,35,76)(21,47,36,75), (2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(22,31)(23,30)(24,29)(25,28)(26,27)(32,42)(33,41)(34,40)(35,39)(36,38)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78) );
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)], [(1,46,37,74),(2,45,38,73),(3,44,39,72),(4,43,40,71),(5,63,41,70),(6,62,42,69),(7,61,22,68),(8,60,23,67),(9,59,24,66),(10,58,25,65),(11,57,26,64),(12,56,27,84),(13,55,28,83),(14,54,29,82),(15,53,30,81),(16,52,31,80),(17,51,32,79),(18,50,33,78),(19,49,34,77),(20,48,35,76),(21,47,36,75)], [(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13),(11,12),(22,31),(23,30),(24,29),(25,28),(26,27),(32,42),(33,41),(34,40),(35,39),(36,38),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78)]])
C21⋊7D4 is a maximal subgroup of
Dic7.D6 Dic3.D14 D7×C3⋊D4 S3×C7⋊D4 D84⋊11C2 D4×D21 D4⋊2D21
C21⋊7D4 is a maximal quotient of C42.4Q8 C2.D84 D4⋊D21 D4.D21 Q8⋊2D21 C21⋊7Q16 C42.38D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | 7B | 7C | 14A | ··· | 14I | 21A | ··· | 21F | 42A | ··· | 42R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 42 | 2 | 42 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D7 | C3⋊D4 | D14 | D21 | C7⋊D4 | D42 | C21⋊7D4 |
| kernel | C21⋊7D4 | Dic21 | D42 | C2×C42 | C2×C14 | C21 | C14 | C2×C6 | C7 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C21⋊7D4 ►in GL2(𝔽43) generated by
| 35 | 31 |
| 31 | 41 |
| 0 | 1 |
| 42 | 0 |
| 36 | 34 |
| 34 | 7 |
G:=sub<GL(2,GF(43))| [35,31,31,41],[0,42,1,0],[36,34,34,7] >;
C21⋊7D4 in GAP, Magma, Sage, TeX
C_{21}\rtimes_7D_4 % in TeX
G:=Group("C21:7D4"); // GroupNames label
G:=SmallGroup(168,38);
// by ID
G=gap.SmallGroup(168,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,61,323,3604]);
// Polycyclic
G:=Group<a,b,c|a^21=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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