direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D21, C6⋊2D14, C14⋊2D6, C21⋊2C23, C42⋊2C22, (C2×C6)⋊3D7, (C2×C14)⋊5S3, (C2×C42)⋊3C2, C7⋊2(C22×S3), C3⋊2(C22×D7), SmallGroup(168,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C22×D21 |
Generators and relations for C22×D21
G = < a,b,c,d | a2=b2=c21=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 372 in 64 conjugacy classes, 31 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C7, C23, D6, C2×C6, D7, C14, C21, C22×S3, D14, C2×C14, D21, C42, C22×D7, D42, C2×C42, C22×D21
Quotients: C1, C2, C22, S3, C23, D6, D7, C22×S3, D14, D21, C22×D7, D42, C22×D21
►On 84 points
Generators in S84
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 61)(10 62)(11 63)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(43 80)(44 81)(45 82)(46 83)(47 84)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 77)(14 76)(15 75)(16 74)(17 73)(18 72)(19 71)(20 70)(21 69)(22 43)(23 63)(24 62)(25 61)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)
G:=sub<Sym(84)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(43,80)(44,81)(45,82)(46,83)(47,84)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,43)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,61)(10,62)(11,63)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(43,80)(44,81)(45,82)(46,83)(47,84)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76)(61,77)(62,78)(63,79), (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), (1,68)(2,67)(3,66)(4,65)(5,64)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,77)(14,76)(15,75)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,43)(23,63)(24,62)(25,61)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44) );
G=PermutationGroup([[(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,61),(10,62),(11,63),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,71),(37,72),(38,73),(39,74),(40,75),(41,76),(42,77)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(43,80),(44,81),(45,82),(46,83),(47,84),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76),(61,77),(62,78),(63,79)], [(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)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,77),(14,76),(15,75),(16,74),(17,73),(18,72),(19,71),(20,70),(21,69),(22,43),(23,63),(24,62),(25,61),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44)]])
C22×D21 is a maximal subgroup of
D42⋊C4 C2.D84 D6⋊D14 C22×S3×D7
C22×D21 is a maximal quotient of D84⋊11C2 D4⋊2D21 Q8⋊3D21
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 7A | 7B | 7C | 14A | ··· | 14I | 21A | ··· | 21F | 42A | ··· | 42R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D7 | D14 | D21 | D42 |
| kernel | C22×D21 | D42 | C2×C42 | C2×C14 | C14 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 6 | 1 | 1 | 3 | 3 | 9 | 6 | 18 |
Matrix representation of C22×D21 ►in GL3(𝔽43) generated by
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 1 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 1 | 0 | 0 |
| 0 | 27 | 18 |
| 0 | 25 | 39 |
| 1 | 0 | 0 |
| 0 | 27 | 18 |
| 0 | 36 | 16 |
G:=sub<GL(3,GF(43))| [42,0,0,0,42,0,0,0,42],[1,0,0,0,42,0,0,0,42],[1,0,0,0,27,25,0,18,39],[1,0,0,0,27,36,0,18,16] >;
C22×D21 in GAP, Magma, Sage, TeX
C_2^2\times D_{21} % in TeX
G:=Group("C2^2xD21"); // GroupNames label
G:=SmallGroup(168,56);
// by ID
G=gap.SmallGroup(168,56);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,323,3604]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^21=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations