direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C6×C7⋊C3, C42⋊4C6, C7⋊2C62, C21⋊6(C2×C6), C14⋊2(C3×C6), (C2×C42)⋊3C3, (C2×C14)⋊4C32, SmallGroup(252,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C6×C7⋊C3 |
Generators and relations for C2×C6×C7⋊C3
G = < a,b,c,d | a2=b6=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Subgroups: 180 in 60 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C3, C3, C22, C6, C6, C7, C32, C2×C6, C2×C6, C14, C3×C6, C7⋊C3, C21, C2×C14, C62, C2×C7⋊C3, C42, C3×C7⋊C3, C22×C7⋊C3, C2×C42, C6×C7⋊C3, C2×C6×C7⋊C3
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C3×C6, C7⋊C3, C62, C2×C7⋊C3, C3×C7⋊C3, C22×C7⋊C3, C6×C7⋊C3, C2×C6×C7⋊C3
►On 84 points
Generators in S84
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)
(1 29 15 22 8 36)(2 30 16 23 9 37)(3 31 17 24 10 38)(4 32 18 25 11 39)(5 33 19 26 12 40)(6 34 20 27 13 41)(7 35 21 28 14 42)(43 71 57 64 50 78)(44 72 58 65 51 79)(45 73 59 66 52 80)(46 74 60 67 53 81)(47 75 61 68 54 82)(48 76 62 69 55 83)(49 77 63 70 56 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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)(37 38 40)(39 42 41)(44 45 47)(46 49 48)(51 52 54)(53 56 55)(58 59 61)(60 63 62)(65 66 68)(67 70 69)(72 73 75)(74 77 76)(79 80 82)(81 84 83)
G:=sub<Sym(84)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84), (1,29,15,22,8,36)(2,30,16,23,9,37)(3,31,17,24,10,38)(4,32,18,25,11,39)(5,33,19,26,12,40)(6,34,20,27,13,41)(7,35,21,28,14,42)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84), (1,29,15,22,8,36)(2,30,16,23,9,37)(3,31,17,24,10,38)(4,32,18,25,11,39)(5,33,19,26,12,40)(6,34,20,27,13,41)(7,35,21,28,14,42)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)(37,38,40)(39,42,41)(44,45,47)(46,49,48)(51,52,54)(53,56,55)(58,59,61)(60,63,62)(65,66,68)(67,70,69)(72,73,75)(74,77,76)(79,80,82)(81,84,83) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84)], [(1,29,15,22,8,36),(2,30,16,23,9,37),(3,31,17,24,10,38),(4,32,18,25,11,39),(5,33,19,26,12,40),(6,34,20,27,13,41),(7,35,21,28,14,42),(43,71,57,64,50,78),(44,72,58,65,51,79),(45,73,59,66,52,80),(46,74,60,67,53,81),(47,75,61,68,54,82),(48,76,62,69,55,83),(49,77,63,70,56,84)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34),(37,38,40),(39,42,41),(44,45,47),(46,49,48),(51,52,54),(53,56,55),(58,59,61),(60,63,62),(65,66,68),(67,70,69),(72,73,75),(74,77,76),(79,80,82),(81,84,83)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 6A | ··· | 6F | 6G | ··· | 6X | 7A | 7B | 14A | ··· | 14F | 21A | 21B | 21C | 21D | 42A | ··· | 42L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 7 | 7 | 14 | ··· | 14 | 21 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C7⋊C3 | C2×C7⋊C3 | C3×C7⋊C3 | C6×C7⋊C3 |
| kernel | C2×C6×C7⋊C3 | C6×C7⋊C3 | C22×C7⋊C3 | C2×C42 | C2×C7⋊C3 | C42 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 6 | 2 | 18 | 6 | 2 | 6 | 4 | 12 |
Matrix representation of C2×C6×C7⋊C3 ►in GL4(𝔽43) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 19 |
| 0 | 0 | 1 | 18 |
| 6 | 0 | 0 | 0 |
| 0 | 1 | 0 | 18 |
| 0 | 0 | 0 | 42 |
| 0 | 0 | 1 | 42 |
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,19,18],[6,0,0,0,0,1,0,0,0,0,0,1,0,18,42,42] >;
C2×C6×C7⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_7\rtimes C_3
% in TeX
G:=Group("C2xC6xC7:C3"); // GroupNames label
G:=SmallGroup(252,38);
// by ID
G=gap.SmallGroup(252,38);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,464]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations