direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3xC7:D4, C21:8D4, D14:5C6, Dic7:4C6, C6.17D14, C42.17C22, C7:5(C3xD4), (C2xC6):1D7, (C2xC42):4C2, (C6xD7):5C2, C2.5(C6xD7), (C2xC14):10C6, C22:2(C3xD7), C14.13(C2xC6), (C3xDic7):4C2, SmallGroup(168,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC7:D4
G = < a,b,c,d | a3=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Quotients: C1, C2, C3, C22, C6, D4, C2xC6, D7, C3xD4, D14, C3xD7, C7:D4, C6xD7, C3xC7:D4
Smallest permutation representation of C3xC7:D4
►On 84 points
Generators in S84
(1 29 15)(2 30 16)(3 31 17)(4 32 18)(5 33 19)(6 34 20)(7 35 21)(8 36 22)(9 37 23)(10 38 24)(11 39 25)(12 40 26)(13 41 27)(14 42 28)(43 71 57)(44 72 58)(45 73 59)(46 74 60)(47 75 61)(48 76 62)(49 77 63)(50 78 64)(51 79 65)(52 80 66)(53 81 67)(54 82 68)(55 83 69)(56 84 70)
(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 43 8 50)(2 49 9 56)(3 48 10 55)(4 47 11 54)(5 46 12 53)(6 45 13 52)(7 44 14 51)(15 57 22 64)(16 63 23 70)(17 62 24 69)(18 61 25 68)(19 60 26 67)(20 59 27 66)(21 58 28 65)(29 71 36 78)(30 77 37 84)(31 76 38 83)(32 75 39 82)(33 74 40 81)(34 73 41 80)(35 72 42 79)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 50)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 64)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 78)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)
G:=sub<Sym(84)| (1,29,15)(2,30,16)(3,31,17)(4,32,18)(5,33,19)(6,34,20)(7,35,21)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (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), (1,43,8,50)(2,49,9,56)(3,48,10,55)(4,47,11,54)(5,46,12,53)(6,45,13,52)(7,44,14,51)(15,57,22,64)(16,63,23,70)(17,62,24,69)(18,61,25,68)(19,60,26,67)(20,59,27,66)(21,58,28,65)(29,71,36,78)(30,77,37,84)(31,76,38,83)(32,75,39,82)(33,74,40,81)(34,73,41,80)(35,72,42,79), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,64)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)>;
G:=Group( (1,29,15)(2,30,16)(3,31,17)(4,32,18)(5,33,19)(6,34,20)(7,35,21)(8,36,22)(9,37,23)(10,38,24)(11,39,25)(12,40,26)(13,41,27)(14,42,28)(43,71,57)(44,72,58)(45,73,59)(46,74,60)(47,75,61)(48,76,62)(49,77,63)(50,78,64)(51,79,65)(52,80,66)(53,81,67)(54,82,68)(55,83,69)(56,84,70), (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), (1,43,8,50)(2,49,9,56)(3,48,10,55)(4,47,11,54)(5,46,12,53)(6,45,13,52)(7,44,14,51)(15,57,22,64)(16,63,23,70)(17,62,24,69)(18,61,25,68)(19,60,26,67)(20,59,27,66)(21,58,28,65)(29,71,36,78)(30,77,37,84)(31,76,38,83)(32,75,39,82)(33,74,40,81)(34,73,41,80)(35,72,42,79), (2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,50)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,64)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,78)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79) );
G=PermutationGroup([[(1,29,15),(2,30,16),(3,31,17),(4,32,18),(5,33,19),(6,34,20),(7,35,21),(8,36,22),(9,37,23),(10,38,24),(11,39,25),(12,40,26),(13,41,27),(14,42,28),(43,71,57),(44,72,58),(45,73,59),(46,74,60),(47,75,61),(48,76,62),(49,77,63),(50,78,64),(51,79,65),(52,80,66),(53,81,67),(54,82,68),(55,83,69),(56,84,70)], [(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)], [(1,43,8,50),(2,49,9,56),(3,48,10,55),(4,47,11,54),(5,46,12,53),(6,45,13,52),(7,44,14,51),(15,57,22,64),(16,63,23,70),(17,62,24,69),(18,61,25,68),(19,60,26,67),(20,59,27,66),(21,58,28,65),(29,71,36,78),(30,77,37,84),(31,76,38,83),(32,75,39,82),(33,74,40,81),(34,73,41,80),(35,72,42,79)], [(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(43,50),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(57,64),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(71,78),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79)]])
C3xC7:D4 is a maximal subgroup of
Dic7.D6 C42.C23 D6:D14 C3xD4xD7
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 12A | 12B | 14A | ··· | 14I | 21A | ··· | 21F | 42A | ··· | 42R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 14 | 1 | 1 | 14 | 1 | 1 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D7 | C3xD4 | D14 | C3xD7 | C7:D4 | C6xD7 | C3xC7:D4 |
| kernel | C3xC7:D4 | C3xDic7 | C6xD7 | C2xC42 | C7:D4 | Dic7 | D14 | C2xC14 | C21 | C2xC6 | C7 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 3 | 2 | 3 | 6 | 6 | 6 | 12 |
Matrix representation of C3xC7:D4 ►in GL2(F43) generated by
| 36 | 0 |
| 0 | 36 |
| 18 | 10 |
| 16 | 40 |
| 0 | 7 |
| 6 | 0 |
| 11 | 21 |
| 25 | 32 |
G:=sub<GL(2,GF(43))| [36,0,0,36],[18,16,10,40],[0,6,7,0],[11,25,21,32] >;
C3xC7:D4 in GAP, Magma, Sage, TeX
C_3\times C_7\rtimes D_4
% in TeX
G:=Group("C3xC7:D4"); // GroupNames label
G:=SmallGroup(168,28);
// by ID
G=gap.SmallGroup(168,28);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-7,141,3604]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^7=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export