direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C7×D4, C28⋊6D6, D12⋊3C14, C84⋊7C22, C42.52C23, C12⋊(C2×C14), C3⋊2(D4×C14), C4⋊1(S3×C14), (C2×C14)⋊7D6, C21⋊15(C2×D4), (C4×S3)⋊1C14, (S3×C28)⋊6C2, (C3×D4)⋊2C14, D6⋊2(C2×C14), (D4×C21)⋊8C2, (C7×D12)⋊9C2, C3⋊D4⋊1C14, (C2×C42)⋊7C22, C22⋊2(S3×C14), (C22×S3)⋊2C14, Dic3⋊1(C2×C14), C6.5(C22×C14), (S3×C14)⋊10C22, C14.42(C22×S3), (C7×Dic3)⋊8C22, (C2×C6)⋊(C2×C14), (S3×C2×C14)⋊6C2, C2.6(S3×C2×C14), (C7×C3⋊D4)⋊5C2, SmallGroup(336,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C7×D4
G = < a,b,c,d,e | a7=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C7, C2×C4, D4, D4, C23, Dic3, C12, D6, D6, D6, C2×C6, C14, C14, C2×D4, C21, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C28, C28, C2×C14, C2×C14, S3×C7, S3×C7, C42, C42, S3×D4, C2×C28, C7×D4, C7×D4, C22×C14, C7×Dic3, C84, S3×C14, S3×C14, S3×C14, C2×C42, D4×C14, S3×C28, C7×D12, C7×C3⋊D4, D4×C21, S3×C2×C14, S3×C7×D4
Quotients: C1, C2, C22, S3, C7, D4, C23, D6, C14, C2×D4, C22×S3, C2×C14, S3×C7, S3×D4, C7×D4, C22×C14, S3×C14, D4×C14, S3×C2×C14, S3×C7×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 43 50)(2 44 51)(3 45 52)(4 46 53)(5 47 54)(6 48 55)(7 49 56)(8 15 78)(9 16 79)(10 17 80)(11 18 81)(12 19 82)(13 20 83)(14 21 84)(22 29 36)(23 30 37)(24 31 38)(25 32 39)(26 33 40)(27 34 41)(28 35 42)(57 64 71)(58 65 72)(59 66 73)(60 67 74)(61 68 75)(62 69 76)(63 70 77)
(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)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 78 57 22)(2 79 58 23)(3 80 59 24)(4 81 60 25)(5 82 61 26)(6 83 62 27)(7 84 63 28)(8 64 29 43)(9 65 30 44)(10 66 31 45)(11 67 32 46)(12 68 33 47)(13 69 34 48)(14 70 35 49)(15 71 36 50)(16 72 37 51)(17 73 38 52)(18 74 39 53)(19 75 40 54)(20 76 41 55)(21 77 42 56)
(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)
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), (1,43,50)(2,44,51)(3,45,52)(4,46,53)(5,47,54)(6,48,55)(7,49,56)(8,15,78)(9,16,79)(10,17,80)(11,18,81)(12,19,82)(13,20,83)(14,21,84)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(57,64,71)(58,65,72)(59,66,73)(60,67,74)(61,68,75)(62,69,76)(63,70,77), (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)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,78,57,22)(2,79,58,23)(3,80,59,24)(4,81,60,25)(5,82,61,26)(6,83,62,27)(7,84,63,28)(8,64,29,43)(9,65,30,44)(10,66,31,45)(11,67,32,46)(12,68,33,47)(13,69,34,48)(14,70,35,49)(15,71,36,50)(16,72,37,51)(17,73,38,52)(18,74,39,53)(19,75,40,54)(20,76,41,55)(21,77,42,56), (8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)>;
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), (1,43,50)(2,44,51)(3,45,52)(4,46,53)(5,47,54)(6,48,55)(7,49,56)(8,15,78)(9,16,79)(10,17,80)(11,18,81)(12,19,82)(13,20,83)(14,21,84)(22,29,36)(23,30,37)(24,31,38)(25,32,39)(26,33,40)(27,34,41)(28,35,42)(57,64,71)(58,65,72)(59,66,73)(60,67,74)(61,68,75)(62,69,76)(63,70,77), (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)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,78,57,22)(2,79,58,23)(3,80,59,24)(4,81,60,25)(5,82,61,26)(6,83,62,27)(7,84,63,28)(8,64,29,43)(9,65,30,44)(10,66,31,45)(11,67,32,46)(12,68,33,47)(13,69,34,48)(14,70,35,49)(15,71,36,50)(16,72,37,51)(17,73,38,52)(18,74,39,53)(19,75,40,54)(20,76,41,55)(21,77,42,56), (8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84) );
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)], [(1,43,50),(2,44,51),(3,45,52),(4,46,53),(5,47,54),(6,48,55),(7,49,56),(8,15,78),(9,16,79),(10,17,80),(11,18,81),(12,19,82),(13,20,83),(14,21,84),(22,29,36),(23,30,37),(24,31,38),(25,32,39),(26,33,40),(27,34,41),(28,35,42),(57,64,71),(58,65,72),(59,66,73),(60,67,74),(61,68,75),(62,69,76),(63,70,77)], [(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),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,78,57,22),(2,79,58,23),(3,80,59,24),(4,81,60,25),(5,82,61,26),(6,83,62,27),(7,84,63,28),(8,64,29,43),(9,65,30,44),(10,66,31,45),(11,67,32,46),(12,68,33,47),(13,69,34,48),(14,70,35,49),(15,71,36,50),(16,72,37,51),(17,73,38,52),(18,74,39,53),(19,75,40,54),(20,76,41,55),(21,77,42,56)], [(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84)]])
105 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 7A | ··· | 7F | 12 | 14A | ··· | 14F | 14G | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 21A | ··· | 21F | 28A | ··· | 28F | 28G | ··· | 28L | 42A | ··· | 42F | 42G | ··· | 42R | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | ··· | 7 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 28 | ··· | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 2 | 2 | 6 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | S3 | D4 | D6 | D6 | S3×C7 | C7×D4 | S3×C14 | S3×C14 | S3×D4 | S3×C7×D4 |
| kernel | S3×C7×D4 | S3×C28 | C7×D12 | C7×C3⋊D4 | D4×C21 | S3×C2×C14 | S3×D4 | C4×S3 | D12 | C3⋊D4 | C3×D4 | C22×S3 | C7×D4 | S3×C7 | C28 | C2×C14 | D4 | S3 | C4 | C22 | C7 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 6 | 6 | 6 | 12 | 6 | 12 | 1 | 2 | 1 | 2 | 6 | 12 | 6 | 12 | 1 | 6 |
Matrix representation of S3×C7×D4 ►in GL4(𝔽337) generated by
| 295 | 0 | 0 | 0 |
| 0 | 295 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 336 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(337))| [295,0,0,0,0,295,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,336,336],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,336,0,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1] >;
S3×C7×D4 in GAP, Magma, Sage, TeX
S_3\times C_7\times D_4
% in TeX
G:=Group("S3xC7xD4"); // GroupNames label
G:=SmallGroup(336,188);
// by ID
G=gap.SmallGroup(336,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-3,548,8069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations