direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7×C3⋊D4, C21⋊9D4, D6⋊2C14, Dic3⋊C14, C14.17D6, C42.22C22, C3⋊2(C7×D4), (C2×C14)⋊3S3, (C2×C42)⋊6C2, (C2×C6)⋊2C14, (S3×C14)⋊5C2, C2.5(S3×C14), C6.5(C2×C14), C22⋊2(S3×C7), (C7×Dic3)⋊4C2, SmallGroup(168,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C3⋊D4
G = < a,b,c,d | a7=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C7×C3⋊D4
►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 62 69)(2 63 70)(3 57 64)(4 58 65)(5 59 66)(6 60 67)(7 61 68)(8 34 41)(9 35 42)(10 29 36)(11 30 37)(12 31 38)(13 32 39)(14 33 40)(15 22 79)(16 23 80)(17 24 81)(18 25 82)(19 26 83)(20 27 84)(21 28 78)(43 54 73)(44 55 74)(45 56 75)(46 50 76)(47 51 77)(48 52 71)(49 53 72)
(1 78 51 10)(2 79 52 11)(3 80 53 12)(4 81 54 13)(5 82 55 14)(6 83 56 8)(7 84 50 9)(15 48 30 70)(16 49 31 64)(17 43 32 65)(18 44 33 66)(19 45 34 67)(20 46 35 68)(21 47 29 69)(22 71 37 63)(23 72 38 57)(24 73 39 58)(25 74 40 59)(26 75 41 60)(27 76 42 61)(28 77 36 62)
(8 83)(9 84)(10 78)(11 79)(12 80)(13 81)(14 82)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 36)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)(28 29)(43 73)(44 74)(45 75)(46 76)(47 77)(48 71)(49 72)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
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), (1,62,69)(2,63,70)(3,57,64)(4,58,65)(5,59,66)(6,60,67)(7,61,68)(8,34,41)(9,35,42)(10,29,36)(11,30,37)(12,31,38)(13,32,39)(14,33,40)(15,22,79)(16,23,80)(17,24,81)(18,25,82)(19,26,83)(20,27,84)(21,28,78)(43,54,73)(44,55,74)(45,56,75)(46,50,76)(47,51,77)(48,52,71)(49,53,72), (1,78,51,10)(2,79,52,11)(3,80,53,12)(4,81,54,13)(5,82,55,14)(6,83,56,8)(7,84,50,9)(15,48,30,70)(16,49,31,64)(17,43,32,65)(18,44,33,66)(19,45,34,67)(20,46,35,68)(21,47,29,69)(22,71,37,63)(23,72,38,57)(24,73,39,58)(25,74,40,59)(26,75,41,60)(27,76,42,61)(28,77,36,62), (8,83)(9,84)(10,78)(11,79)(12,80)(13,81)(14,82)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,36)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(28,29)(43,73)(44,74)(45,75)(46,76)(47,77)(48,71)(49,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)>;
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), (1,62,69)(2,63,70)(3,57,64)(4,58,65)(5,59,66)(6,60,67)(7,61,68)(8,34,41)(9,35,42)(10,29,36)(11,30,37)(12,31,38)(13,32,39)(14,33,40)(15,22,79)(16,23,80)(17,24,81)(18,25,82)(19,26,83)(20,27,84)(21,28,78)(43,54,73)(44,55,74)(45,56,75)(46,50,76)(47,51,77)(48,52,71)(49,53,72), (1,78,51,10)(2,79,52,11)(3,80,53,12)(4,81,54,13)(5,82,55,14)(6,83,56,8)(7,84,50,9)(15,48,30,70)(16,49,31,64)(17,43,32,65)(18,44,33,66)(19,45,34,67)(20,46,35,68)(21,47,29,69)(22,71,37,63)(23,72,38,57)(24,73,39,58)(25,74,40,59)(26,75,41,60)(27,76,42,61)(28,77,36,62), (8,83)(9,84)(10,78)(11,79)(12,80)(13,81)(14,82)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,36)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(28,29)(43,73)(44,74)(45,75)(46,76)(47,77)(48,71)(49,72)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70) );
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)], [(1,62,69),(2,63,70),(3,57,64),(4,58,65),(5,59,66),(6,60,67),(7,61,68),(8,34,41),(9,35,42),(10,29,36),(11,30,37),(12,31,38),(13,32,39),(14,33,40),(15,22,79),(16,23,80),(17,24,81),(18,25,82),(19,26,83),(20,27,84),(21,28,78),(43,54,73),(44,55,74),(45,56,75),(46,50,76),(47,51,77),(48,52,71),(49,53,72)], [(1,78,51,10),(2,79,52,11),(3,80,53,12),(4,81,54,13),(5,82,55,14),(6,83,56,8),(7,84,50,9),(15,48,30,70),(16,49,31,64),(17,43,32,65),(18,44,33,66),(19,45,34,67),(20,46,35,68),(21,47,29,69),(22,71,37,63),(23,72,38,57),(24,73,39,58),(25,74,40,59),(26,75,41,60),(27,76,42,61),(28,77,36,62)], [(8,83),(9,84),(10,78),(11,79),(12,80),(13,81),(14,82),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,36),(22,30),(23,31),(24,32),(25,33),(26,34),(27,35),(28,29),(43,73),(44,74),(45,75),(46,76),(47,77),(48,71),(49,72),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)]])
C7×C3⋊D4 is a maximal subgroup of
C42.C23 Dic3.D14 D6⋊D14 S3×C7×D4
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 21A | ··· | 21F | 28A | ··· | 28F | 42A | ··· | 42R |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
63 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 | C7 | C14 | C14 | C14 | S3 | D4 | D6 | C3⋊D4 | S3×C7 | C7×D4 | S3×C14 | C7×C3⋊D4 |
| kernel | C7×C3⋊D4 | C7×Dic3 | S3×C14 | C2×C42 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C14 | C21 | C14 | C7 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 2 | 6 | 6 | 6 | 12 |
Matrix representation of C7×C3⋊D4 ►in GL2(𝔽43) generated by
| 41 | 0 |
| 0 | 41 |
| 41 | 7 |
| 18 | 1 |
| 13 | 27 |
| 16 | 30 |
| 1 | 36 |
| 0 | 42 |
G:=sub<GL(2,GF(43))| [41,0,0,41],[41,18,7,1],[13,16,27,30],[1,0,36,42] >;
C7×C3⋊D4 in GAP, Magma, Sage, TeX
C_7\times C_3\rtimes D_4
% in TeX
G:=Group("C7xC3:D4"); // GroupNames label
G:=SmallGroup(168,33);
// by ID
G=gap.SmallGroup(168,33);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-3,301,2804]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=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
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