direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C7⋊D4, D14⋊3D6, D6⋊7D14, Dic7⋊1D6, D42⋊4C22, C42.26C23, Dic21⋊2C22, C7⋊5(S3×D4), C21⋊8(C2×D4), (S3×C7)⋊2D4, (C2×C14)⋊8D6, (C2×C6)⋊1D14, C7⋊D12⋊5C2, C21⋊D4⋊6C2, C21⋊7D4⋊6C2, C22⋊2(S3×D7), (C2×C42)⋊3C22, (S3×Dic7)⋊5C2, (C22×S3)⋊3D7, (C6×D7)⋊3C22, (S3×C14)⋊7C22, C6.26(C22×D7), C14.26(C22×S3), (C3×Dic7)⋊1C22, (C2×S3×D7)⋊5C2, C3⋊2(C2×C7⋊D4), (S3×C2×C14)⋊3C2, C2.26(C2×S3×D7), (C3×C7⋊D4)⋊3C2, SmallGroup(336,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C7⋊D4
G = < a,b,c,d,e | a3=b2=c7=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 636 in 108 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, S3, C6, C6, C7, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, D7, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C22×S3, Dic7, Dic7, D14, D14, C2×C14, C2×C14, S3×C7, S3×C7, C3×D7, D21, C42, C42, S3×D4, C2×Dic7, C7⋊D4, C7⋊D4, C22×D7, C22×C14, C3×Dic7, Dic21, S3×D7, C6×D7, S3×C14, S3×C14, D42, C2×C42, C2×C7⋊D4, S3×Dic7, C21⋊D4, C7⋊D12, C3×C7⋊D4, C21⋊7D4, C2×S3×D7, S3×C2×C14, S3×C7⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, C22×S3, D14, S3×D4, C7⋊D4, C22×D7, S3×D7, C2×C7⋊D4, C2×S3×D7, S3×C7⋊D4
►On 84 points
Generators in S84
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(43 50 57)(44 51 58)(45 52 59)(46 53 60)(47 54 61)(48 55 62)(49 56 63)(64 71 78)(65 72 79)(66 73 80)(67 74 81)(68 75 82)(69 76 83)(70 77 84)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 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 43 22 64)(2 49 23 70)(3 48 24 69)(4 47 25 68)(5 46 26 67)(6 45 27 66)(7 44 28 65)(8 50 29 71)(9 56 30 77)(10 55 31 76)(11 54 32 75)(12 53 33 74)(13 52 34 73)(14 51 35 72)(15 57 36 78)(16 63 37 84)(17 62 38 83)(18 61 39 82)(19 60 40 81)(20 59 41 80)(21 58 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 64)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 71)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 78)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)
G:=sub<Sym(84)| (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,50,57)(44,51,58)(45,52,59)(46,53,60)(47,54,61)(48,55,62)(49,56,63)(64,71,78)(65,72,79)(66,73,80)(67,74,81)(68,75,82)(69,76,83)(70,77,84), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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), (1,43,22,64)(2,49,23,70)(3,48,24,69)(4,47,25,68)(5,46,26,67)(6,45,27,66)(7,44,28,65)(8,50,29,71)(9,56,30,77)(10,55,31,76)(11,54,32,75)(12,53,33,74)(13,52,34,73)(14,51,35,72)(15,57,36,78)(16,63,37,84)(17,62,38,83)(18,61,39,82)(19,60,40,81)(20,59,41,80)(21,58,42,79), (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)(43,64)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,71)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)>;
G:=Group( (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(43,50,57)(44,51,58)(45,52,59)(46,53,60)(47,54,61)(48,55,62)(49,56,63)(64,71,78)(65,72,79)(66,73,80)(67,74,81)(68,75,82)(69,76,83)(70,77,84), (8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (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), (1,43,22,64)(2,49,23,70)(3,48,24,69)(4,47,25,68)(5,46,26,67)(6,45,27,66)(7,44,28,65)(8,50,29,71)(9,56,30,77)(10,55,31,76)(11,54,32,75)(12,53,33,74)(13,52,34,73)(14,51,35,72)(15,57,36,78)(16,63,37,84)(17,62,38,83)(18,61,39,82)(19,60,40,81)(20,59,41,80)(21,58,42,79), (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)(43,64)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,71)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79) );
G=PermutationGroup([[(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42),(43,50,57),(44,51,58),(45,52,59),(46,53,60),(47,54,61),(48,55,62),(49,56,63),(64,71,78),(65,72,79),(66,73,80),(67,74,81),(68,75,82),(69,76,83),(70,77,84)], [(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(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)], [(1,43,22,64),(2,49,23,70),(3,48,24,69),(4,47,25,68),(5,46,26,67),(6,45,27,66),(7,44,28,65),(8,50,29,71),(9,56,30,77),(10,55,31,76),(11,54,32,75),(12,53,33,74),(13,52,34,73),(14,51,35,72),(15,57,36,78),(16,63,37,84),(17,62,38,83),(18,61,39,82),(19,60,40,81),(20,59,41,80),(21,58,42,79)], [(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),(43,64),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,71),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,78),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79)]])
51 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 12 | 14A | ··· | 14I | 14J | ··· | 14U | 21A | 21B | 21C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 14 | 42 | 2 | 14 | 42 | 2 | 4 | 28 | 2 | 2 | 2 | 28 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D7 | D14 | D14 | C7⋊D4 | S3×D4 | S3×D7 | C2×S3×D7 | S3×C7⋊D4 |
| kernel | S3×C7⋊D4 | S3×Dic7 | C21⋊D4 | C7⋊D12 | C3×C7⋊D4 | C21⋊7D4 | C2×S3×D7 | S3×C2×C14 | C7⋊D4 | S3×C7 | Dic7 | D14 | C2×C14 | C22×S3 | D6 | C2×C6 | S3 | C7 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 6 | 3 | 12 | 1 | 3 | 3 | 6 |
Matrix representation of S3×C7⋊D4 ►in GL6(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 336 | 336 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 336 | 336 |
| 303 | 336 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 303 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 295 | 21 | 0 | 0 |
| 0 | 0 | 269 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 303 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 336 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,1,0,0,0,0,336,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,336,0,0,0,0,0,336],[303,1,0,0,0,0,336,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,303,0,0,0,0,0,336,0,0,0,0,0,0,295,269,0,0,0,0,21,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,303,0,0,0,0,0,336,0,0,0,0,0,0,1,4,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
S3×C7⋊D4 in GAP, Magma, Sage, TeX
S_3\times C_7\rtimes D_4
% in TeX
G:=Group("S3xC7:D4"); // GroupNames label
G:=SmallGroup(336,162);
// by ID
G=gap.SmallGroup(336,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,218,490,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^7=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations