direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4xD21, C84:2C2, C28:2S3, C12:2D7, C6.9D14, C2.1D42, C14.9D6, D42.2C2, Dic21:5C2, C42.9C22, C7:2(C4xS3), C3:2(C4xD7), C21:4(C2xC4), SmallGroup(168,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C4xD21 |
Generators and relations for C4xD21
G = < a,b,c | a4=b21=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, S3, C2xC4, D6, D7, C4xS3, D14, D21, C4xD7, D42, C4xD21
Smallest permutation representation of C4xD21
►On 84 points
Generators in S84
(1 76 30 56)(2 77 31 57)(3 78 32 58)(4 79 33 59)(5 80 34 60)(6 81 35 61)(7 82 36 62)(8 83 37 63)(9 84 38 43)(10 64 39 44)(11 65 40 45)(12 66 41 46)(13 67 42 47)(14 68 22 48)(15 69 23 49)(16 70 24 50)(17 71 25 51)(18 72 26 52)(19 73 27 53)(20 74 28 54)(21 75 29 55)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(43 67)(44 66)(45 65)(46 64)(47 84)(48 83)(49 82)(50 81)(51 80)(52 79)(53 78)(54 77)(55 76)(56 75)(57 74)(58 73)(59 72)(60 71)(61 70)(62 69)(63 68)
G:=sub<Sym(84)| (1,76,30,56)(2,77,31,57)(3,78,32,58)(4,79,33,59)(5,80,34,60)(6,81,35,61)(7,82,36,62)(8,83,37,63)(9,84,38,43)(10,64,39,44)(11,65,40,45)(12,66,41,46)(13,67,42,47)(14,68,22,48)(15,69,23,49)(16,70,24,50)(17,71,25,51)(18,72,26,52)(19,73,27,53)(20,74,28,54)(21,75,29,55), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(43,67)(44,66)(45,65)(46,64)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68)>;
G:=Group( (1,76,30,56)(2,77,31,57)(3,78,32,58)(4,79,33,59)(5,80,34,60)(6,81,35,61)(7,82,36,62)(8,83,37,63)(9,84,38,43)(10,64,39,44)(11,65,40,45)(12,66,41,46)(13,67,42,47)(14,68,22,48)(15,69,23,49)(16,70,24,50)(17,71,25,51)(18,72,26,52)(19,73,27,53)(20,74,28,54)(21,75,29,55), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(43,67)(44,66)(45,65)(46,64)(47,84)(48,83)(49,82)(50,81)(51,80)(52,79)(53,78)(54,77)(55,76)(56,75)(57,74)(58,73)(59,72)(60,71)(61,70)(62,69)(63,68) );
G=PermutationGroup([[(1,76,30,56),(2,77,31,57),(3,78,32,58),(4,79,33,59),(5,80,34,60),(6,81,35,61),(7,82,36,62),(8,83,37,63),(9,84,38,43),(10,64,39,44),(11,65,40,45),(12,66,41,46),(13,67,42,47),(14,68,22,48),(15,69,23,49),(16,70,24,50),(17,71,25,51),(18,72,26,52),(19,73,27,53),(20,74,28,54),(21,75,29,55)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(43,67),(44,66),(45,65),(46,64),(47,84),(48,83),(49,82),(50,81),(51,80),(52,79),(53,78),(54,77),(55,76),(56,75),(57,74),(58,73),(59,72),(60,71),(61,70),(62,69),(63,68)]])
C4xD21 is a maximal subgroup of
D21:C8 D42.C4 C56:S3 D28:S3 D12:D7 D21:Q8 D6.D14 C4xS3xD7 C28:D6 D84:11C2 D4:2D21 Q8:3D21
C4xD21 is a maximal quotient of
C56:S3 C42.4Q8 C2.D84
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6 | 7A | 7B | 7C | 12A | 12B | 14A | 14B | 14C | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42F | 84A | ··· | 84L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 7 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 21 | 21 | 2 | 1 | 1 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | D7 | C4xS3 | D14 | D21 | C4xD7 | D42 | C4xD21 |
| kernel | C4xD21 | Dic21 | C84 | D42 | D21 | C28 | C14 | C12 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C4xD21 ►in GL2(F41) generated by
| 32 | 0 |
| 0 | 32 |
| 17 | 8 |
| 8 | 40 |
| 1 | 0 |
| 8 | 40 |
G:=sub<GL(2,GF(41))| [32,0,0,32],[17,8,8,40],[1,8,0,40] >;
C4xD21 in GAP, Magma, Sage, TeX
C_4\times D_{21} % in TeX
G:=Group("C4xD21"); // GroupNames label
G:=SmallGroup(168,35);
// by ID
G=gap.SmallGroup(168,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-7,26,323,3604]);
// Polycyclic
G:=Group<a,b,c|a^4=b^21=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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