direct product, metabelian, soluble, monomial, A-group
Aliases: C7×C42⋊C3, C42⋊C21, (C4×C28)⋊1C3, C22.(C7×A4), (C2×C14).1A4, SmallGroup(336,56)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C7×C42⋊C3 |
Generators and relations for C7×C42⋊C3
G = < a,b,c,d | a7=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Smallest permutation representation of C7×C42⋊C3
►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 30 39 43)(2 31 40 44)(3 32 41 45)(4 33 42 46)(5 34 36 47)(6 35 37 48)(7 29 38 49)(8 21)(9 15)(10 16)(11 17)(12 18)(13 19)(14 20)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(50 63 71 66)(51 57 72 67)(52 58 73 68)(53 59 74 69)(54 60 75 70)(55 61 76 64)(56 62 77 65)
(1 43 39 30)(2 44 40 31)(3 45 41 32)(4 46 42 33)(5 47 36 34)(6 48 37 35)(7 49 38 29)(8 23 21 84)(9 24 15 78)(10 25 16 79)(11 26 17 80)(12 27 18 81)(13 28 19 82)(14 22 20 83)
(1 15 66)(2 16 67)(3 17 68)(4 18 69)(5 19 70)(6 20 64)(7 21 65)(8 62 38)(9 63 39)(10 57 40)(11 58 41)(12 59 42)(13 60 36)(14 61 37)(22 76 48)(23 77 49)(24 71 43)(25 72 44)(26 73 45)(27 74 46)(28 75 47)(29 84 56)(30 78 50)(31 79 51)(32 80 52)(33 81 53)(34 82 54)(35 83 55)
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), (1,30,39,43)(2,31,40,44)(3,32,41,45)(4,33,42,46)(5,34,36,47)(6,35,37,48)(7,29,38,49)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(50,63,71,66)(51,57,72,67)(52,58,73,68)(53,59,74,69)(54,60,75,70)(55,61,76,64)(56,62,77,65), (1,43,39,30)(2,44,40,31)(3,45,41,32)(4,46,42,33)(5,47,36,34)(6,48,37,35)(7,49,38,29)(8,23,21,84)(9,24,15,78)(10,25,16,79)(11,26,17,80)(12,27,18,81)(13,28,19,82)(14,22,20,83), (1,15,66)(2,16,67)(3,17,68)(4,18,69)(5,19,70)(6,20,64)(7,21,65)(8,62,38)(9,63,39)(10,57,40)(11,58,41)(12,59,42)(13,60,36)(14,61,37)(22,76,48)(23,77,49)(24,71,43)(25,72,44)(26,73,45)(27,74,46)(28,75,47)(29,84,56)(30,78,50)(31,79,51)(32,80,52)(33,81,53)(34,82,54)(35,83,55)>;
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), (1,30,39,43)(2,31,40,44)(3,32,41,45)(4,33,42,46)(5,34,36,47)(6,35,37,48)(7,29,38,49)(8,21)(9,15)(10,16)(11,17)(12,18)(13,19)(14,20)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(50,63,71,66)(51,57,72,67)(52,58,73,68)(53,59,74,69)(54,60,75,70)(55,61,76,64)(56,62,77,65), (1,43,39,30)(2,44,40,31)(3,45,41,32)(4,46,42,33)(5,47,36,34)(6,48,37,35)(7,49,38,29)(8,23,21,84)(9,24,15,78)(10,25,16,79)(11,26,17,80)(12,27,18,81)(13,28,19,82)(14,22,20,83), (1,15,66)(2,16,67)(3,17,68)(4,18,69)(5,19,70)(6,20,64)(7,21,65)(8,62,38)(9,63,39)(10,57,40)(11,58,41)(12,59,42)(13,60,36)(14,61,37)(22,76,48)(23,77,49)(24,71,43)(25,72,44)(26,73,45)(27,74,46)(28,75,47)(29,84,56)(30,78,50)(31,79,51)(32,80,52)(33,81,53)(34,82,54)(35,83,55) );
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)], [(1,30,39,43),(2,31,40,44),(3,32,41,45),(4,33,42,46),(5,34,36,47),(6,35,37,48),(7,29,38,49),(8,21),(9,15),(10,16),(11,17),(12,18),(13,19),(14,20),(22,83),(23,84),(24,78),(25,79),(26,80),(27,81),(28,82),(50,63,71,66),(51,57,72,67),(52,58,73,68),(53,59,74,69),(54,60,75,70),(55,61,76,64),(56,62,77,65)], [(1,43,39,30),(2,44,40,31),(3,45,41,32),(4,46,42,33),(5,47,36,34),(6,48,37,35),(7,49,38,29),(8,23,21,84),(9,24,15,78),(10,25,16,79),(11,26,17,80),(12,27,18,81),(13,28,19,82),(14,22,20,83)], [(1,15,66),(2,16,67),(3,17,68),(4,18,69),(5,19,70),(6,20,64),(7,21,65),(8,62,38),(9,63,39),(10,57,40),(11,58,41),(12,59,42),(13,60,36),(14,61,37),(22,76,48),(23,77,49),(24,71,43),(25,72,44),(26,73,45),(27,74,46),(28,75,47),(29,84,56),(30,78,50),(31,79,51),(32,80,52),(33,81,53),(34,82,54),(35,83,55)]])
56 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 28A | ··· | 28X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 |
| size | 1 | 3 | 16 | 16 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 16 | ··· | 16 | 3 | ··· | 3 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||
| image | C1 | C3 | C7 | C21 | A4 | C42⋊C3 | C7×A4 | C7×C42⋊C3 |
| kernel | C7×C42⋊C3 | C4×C28 | C42⋊C3 | C42 | C2×C14 | C7 | C22 | C1 |
| # reps | 1 | 2 | 6 | 12 | 1 | 4 | 6 | 24 |
Matrix representation of C7×C42⋊C3 ►in GL3(𝔽337) generated by
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 148 | 0 | 0 |
| 0 | 148 | 0 |
| 331 | 0 | 336 |
| 189 | 0 | 0 |
| 237 | 1 | 0 |
| 220 | 0 | 148 |
| 128 | 335 | 0 |
| 0 | 209 | 1 |
| 0 | 129 | 0 |
G:=sub<GL(3,GF(337))| [64,0,0,0,64,0,0,0,64],[148,0,331,0,148,0,0,0,336],[189,237,220,0,1,0,0,0,148],[128,0,0,335,209,129,0,1,0] >;
C7×C42⋊C3 in GAP, Magma, Sage, TeX
C_7\times C_4^2\rtimes C_3
% in TeX
G:=Group("C7xC4^2:C3"); // GroupNames label
G:=SmallGroup(336,56);
// by ID
G=gap.SmallGroup(336,56);
# by ID
G:=PCGroup([6,-3,-7,-2,2,-2,2,758,230,5547,69,5044,9077]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
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