direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×D12, C28⋊1D6, D6⋊1D14, C12⋊4D14, D84⋊10C2, C84⋊3C22, Dic7⋊3D6, D14.10D6, D42⋊1C22, C42.12C23, C3⋊1(D4×D7), C4⋊2(S3×D7), C21⋊1(C2×D4), C7⋊1(C2×D12), (C4×D7)⋊3S3, (C3×D7)⋊1D4, (C7×D12)⋊3C2, (C12×D7)⋊3C2, C7⋊D12⋊3C2, (S3×C14)⋊1C22, C6.12(C22×D7), C14.12(C22×S3), (C3×Dic7)⋊4C22, (C6×D7).10C22, (C2×S3×D7)⋊1C2, C2.15(C2×S3×D7), SmallGroup(336,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×D12
G = < a,b,c,d | a7=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 788 in 108 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C7, C2×C4, D4, C23, C12, C12, D6, D6, C2×C6, D7, D7, C14, C14, C2×D4, C21, D12, D12, C2×C12, C22×S3, Dic7, C28, D14, D14, C2×C14, S3×C7, C3×D7, D21, C42, C2×D12, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C3×Dic7, C84, S3×D7, C6×D7, S3×C14, D42, D4×D7, C7⋊D12, C12×D7, C7×D12, D84, C2×S3×D7, D7×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, D7, C2×D4, D12, C22×S3, D14, C2×D12, C22×D7, S3×D7, D4×D7, C2×S3×D7, D7×D12
►On 84 points
Generators in S84
(1 53 68 77 18 37 30)(2 54 69 78 19 38 31)(3 55 70 79 20 39 32)(4 56 71 80 21 40 33)(5 57 72 81 22 41 34)(6 58 61 82 23 42 35)(7 59 62 83 24 43 36)(8 60 63 84 13 44 25)(9 49 64 73 14 45 26)(10 50 65 74 15 46 27)(11 51 66 75 16 47 28)(12 52 67 76 17 48 29)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 69)(14 70)(15 71)(16 72)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(73 79)(74 80)(75 81)(76 82)(77 83)(78 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 80)(74 79)(75 78)(76 77)(81 84)(82 83)
G:=sub<Sym(84)| (1,53,68,77,18,37,30)(2,54,69,78,19,38,31)(3,55,70,79,20,39,32)(4,56,71,80,21,40,33)(5,57,72,81,22,41,34)(6,58,61,82,23,42,35)(7,59,62,83,24,43,36)(8,60,63,84,13,44,25)(9,49,64,73,14,45,26)(10,50,65,74,15,46,27)(11,51,66,75,16,47,28)(12,52,67,76,17,48,29), (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,69)(14,70)(15,71)(16,72)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83)>;
G:=Group( (1,53,68,77,18,37,30)(2,54,69,78,19,38,31)(3,55,70,79,20,39,32)(4,56,71,80,21,40,33)(5,57,72,81,22,41,34)(6,58,61,82,23,42,35)(7,59,62,83,24,43,36)(8,60,63,84,13,44,25)(9,49,64,73,14,45,26)(10,50,65,74,15,46,27)(11,51,66,75,16,47,28)(12,52,67,76,17,48,29), (1,36)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,69)(14,70)(15,71)(16,72)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,34)(26,33)(27,32)(28,31)(29,30)(35,36)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,62)(63,72)(64,71)(65,70)(66,69)(67,68)(73,80)(74,79)(75,78)(76,77)(81,84)(82,83) );
G=PermutationGroup([[(1,53,68,77,18,37,30),(2,54,69,78,19,38,31),(3,55,70,79,20,39,32),(4,56,71,80,21,40,33),(5,57,72,81,22,41,34),(6,58,61,82,23,42,35),(7,59,62,83,24,43,36),(8,60,63,84,13,44,25),(9,49,64,73,14,45,26),(10,50,65,74,15,46,27),(11,51,66,75,16,47,28),(12,52,67,76,17,48,29)], [(1,36),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,69),(14,70),(15,71),(16,72),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,34),(26,33),(27,32),(28,31),(29,30),(35,36),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,62),(63,72),(64,71),(65,70),(66,69),(67,68),(73,80),(74,79),(75,78),(76,77),(81,84),(82,83)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 14D | ··· | 14I | 21A | 21B | 21C | 28A | 28B | 28C | 42A | 42B | 42C | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 28 | 28 | 28 | 42 | 42 | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 6 | 6 | 7 | 7 | 42 | 42 | 2 | 2 | 14 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
45 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | D7 | D12 | D14 | D14 | S3×D7 | D4×D7 | C2×S3×D7 | D7×D12 |
| kernel | D7×D12 | C7⋊D12 | C12×D7 | C7×D12 | D84 | C2×S3×D7 | C4×D7 | C3×D7 | Dic7 | C28 | D14 | D12 | D7 | C12 | D6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 3 | 6 | 3 | 3 | 3 | 6 |
Matrix representation of D7×D12 ►in GL4(𝔽337) generated by
| 109 | 1 | 0 | 0 |
| 251 | 194 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 194 | 303 | 0 | 0 |
| 86 | 143 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 292 | 244 |
| 0 | 0 | 29 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 244 |
| 0 | 0 | 308 | 0 |
G:=sub<GL(4,GF(337))| [109,251,0,0,1,194,0,0,0,0,1,0,0,0,0,1],[194,86,0,0,303,143,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,292,29,0,0,244,0],[1,0,0,0,0,1,0,0,0,0,0,308,0,0,244,0] >;
D7×D12 in GAP, Magma, Sage, TeX
D_7\times D_{12} % in TeX
G:=Group("D7xD12"); // GroupNames label
G:=SmallGroup(336,148);
// by ID
G=gap.SmallGroup(336,148);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,116,50,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations