metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊11D4, C42.167D6, C6.782+ (1+4), C4⋊1D4⋊8S3, C4.72(S3×D4), (C4×D12)⋊50C2, C12⋊5(C4○D4), C3⋊4(Q8⋊6D4), C12.67(C2×D4), D6⋊3D4⋊37C2, C12⋊3D4⋊27C2, C4⋊1(D4⋊2S3), (D4×Dic3)⋊35C2, (C4×Dic6)⋊51C2, (C2×D4).115D6, C6.96(C22×D4), (C2×C6).262C24, Dic3.29(C2×D4), C23.14D6⋊37C2, C2.82(D4⋊6D6), (C2×C12).636C23, (C4×C12).204C22, D6⋊C4.149C22, (C6×D4).214C22, (C22×C6).76C23, C23.78(C22×S3), (C2×D12).270C22, C4⋊Dic3.381C22, C22.283(S3×C23), Dic3⋊C4.164C22, (C22×S3).116C23, (C4×Dic3).155C22, (C2×Dic3).269C23, (C2×Dic6).301C22, C6.D4.73C22, (C22×Dic3).158C22, C2.69(C2×S3×D4), (C3×C4⋊1D4)⋊9C2, C6.97(C2×C4○D4), (C2×D4⋊2S3)⋊22C2, C2.61(C2×D4⋊2S3), (S3×C2×C4).139C22, (C2×C4).598(C22×S3), (C2×C3⋊D4).78C22, SmallGroup(192,1277)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 832 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×9], C22, C22 [×18], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×18], D4 [×24], Q8 [×4], C23 [×4], C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×4], C12, D6 [×6], C2×C6, C2×C6 [×12], C42, C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], C2×D4 [×9], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×8], C3⋊D4 [×12], C2×C12 [×3], C3×D4 [×10], C22×S3 [×2], C22×C6 [×4], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4, C4⋊1D4 [×2], C2×C4○D4 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C2×Dic6, S3×C2×C4 [×2], C2×D12, D4⋊2S3 [×8], C22×Dic3 [×4], C2×C3⋊D4 [×8], C6×D4 [×6], Q8⋊6D4, C4×Dic6, C4×D12, D4×Dic3 [×2], D6⋊3D4 [×2], C23.14D6 [×4], C12⋊3D4 [×2], C3×C4⋊1D4, C2×D4⋊2S3 [×2], Dic6⋊11D4
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), S3×D4 [×2], D4⋊2S3 [×2], S3×C23, Q8⋊6D4, C2×S3×D4, C2×D4⋊2S3, D4⋊6D6, Dic6⋊11D4
Generators and relations
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a6b, dcd=c-1 >
►On 96 points
(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)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 23 7 17)(2 22 8 16)(3 21 9 15)(4 20 10 14)(5 19 11 13)(6 18 12 24)(25 49 31 55)(26 60 32 54)(27 59 33 53)(28 58 34 52)(29 57 35 51)(30 56 36 50)(37 88 43 94)(38 87 44 93)(39 86 45 92)(40 85 46 91)(41 96 47 90)(42 95 48 89)(61 84 67 78)(62 83 68 77)(63 82 69 76)(64 81 70 75)(65 80 71 74)(66 79 72 73)
(1 57 93 77)(2 58 94 78)(3 59 95 79)(4 60 96 80)(5 49 85 81)(6 50 86 82)(7 51 87 83)(8 52 88 84)(9 53 89 73)(10 54 90 74)(11 55 91 75)(12 56 92 76)(13 25 40 64)(14 26 41 65)(15 27 42 66)(16 28 43 67)(17 29 44 68)(18 30 45 69)(19 31 46 70)(20 32 47 71)(21 33 48 72)(22 34 37 61)(23 35 38 62)(24 36 39 63)
(1 62)(2 69)(3 64)(4 71)(5 66)(6 61)(7 68)(8 63)(9 70)(10 65)(11 72)(12 67)(13 79)(14 74)(15 81)(16 76)(17 83)(18 78)(19 73)(20 80)(21 75)(22 82)(23 77)(24 84)(25 95)(26 90)(27 85)(28 92)(29 87)(30 94)(31 89)(32 96)(33 91)(34 86)(35 93)(36 88)(37 50)(38 57)(39 52)(40 59)(41 54)(42 49)(43 56)(44 51)(45 58)(46 53)(47 60)(48 55)
G:=sub<Sym(96)| (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)(85,86,87,88,89,90,91,92,93,94,95,96), (1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(61,84,67,78)(62,83,68,77)(63,82,69,76)(64,81,70,75)(65,80,71,74)(66,79,72,73), (1,57,93,77)(2,58,94,78)(3,59,95,79)(4,60,96,80)(5,49,85,81)(6,50,86,82)(7,51,87,83)(8,52,88,84)(9,53,89,73)(10,54,90,74)(11,55,91,75)(12,56,92,76)(13,25,40,64)(14,26,41,65)(15,27,42,66)(16,28,43,67)(17,29,44,68)(18,30,45,69)(19,31,46,70)(20,32,47,71)(21,33,48,72)(22,34,37,61)(23,35,38,62)(24,36,39,63), (1,62)(2,69)(3,64)(4,71)(5,66)(6,61)(7,68)(8,63)(9,70)(10,65)(11,72)(12,67)(13,79)(14,74)(15,81)(16,76)(17,83)(18,78)(19,73)(20,80)(21,75)(22,82)(23,77)(24,84)(25,95)(26,90)(27,85)(28,92)(29,87)(30,94)(31,89)(32,96)(33,91)(34,86)(35,93)(36,88)(37,50)(38,57)(39,52)(40,59)(41,54)(42,49)(43,56)(44,51)(45,58)(46,53)(47,60)(48,55)>;
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)(85,86,87,88,89,90,91,92,93,94,95,96), (1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24)(25,49,31,55)(26,60,32,54)(27,59,33,53)(28,58,34,52)(29,57,35,51)(30,56,36,50)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(61,84,67,78)(62,83,68,77)(63,82,69,76)(64,81,70,75)(65,80,71,74)(66,79,72,73), (1,57,93,77)(2,58,94,78)(3,59,95,79)(4,60,96,80)(5,49,85,81)(6,50,86,82)(7,51,87,83)(8,52,88,84)(9,53,89,73)(10,54,90,74)(11,55,91,75)(12,56,92,76)(13,25,40,64)(14,26,41,65)(15,27,42,66)(16,28,43,67)(17,29,44,68)(18,30,45,69)(19,31,46,70)(20,32,47,71)(21,33,48,72)(22,34,37,61)(23,35,38,62)(24,36,39,63), (1,62)(2,69)(3,64)(4,71)(5,66)(6,61)(7,68)(8,63)(9,70)(10,65)(11,72)(12,67)(13,79)(14,74)(15,81)(16,76)(17,83)(18,78)(19,73)(20,80)(21,75)(22,82)(23,77)(24,84)(25,95)(26,90)(27,85)(28,92)(29,87)(30,94)(31,89)(32,96)(33,91)(34,86)(35,93)(36,88)(37,50)(38,57)(39,52)(40,59)(41,54)(42,49)(43,56)(44,51)(45,58)(46,53)(47,60)(48,55) );
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),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,23,7,17),(2,22,8,16),(3,21,9,15),(4,20,10,14),(5,19,11,13),(6,18,12,24),(25,49,31,55),(26,60,32,54),(27,59,33,53),(28,58,34,52),(29,57,35,51),(30,56,36,50),(37,88,43,94),(38,87,44,93),(39,86,45,92),(40,85,46,91),(41,96,47,90),(42,95,48,89),(61,84,67,78),(62,83,68,77),(63,82,69,76),(64,81,70,75),(65,80,71,74),(66,79,72,73)], [(1,57,93,77),(2,58,94,78),(3,59,95,79),(4,60,96,80),(5,49,85,81),(6,50,86,82),(7,51,87,83),(8,52,88,84),(9,53,89,73),(10,54,90,74),(11,55,91,75),(12,56,92,76),(13,25,40,64),(14,26,41,65),(15,27,42,66),(16,28,43,67),(17,29,44,68),(18,30,45,69),(19,31,46,70),(20,32,47,71),(21,33,48,72),(22,34,37,61),(23,35,38,62),(24,36,39,63)], [(1,62),(2,69),(3,64),(4,71),(5,66),(6,61),(7,68),(8,63),(9,70),(10,65),(11,72),(12,67),(13,79),(14,74),(15,81),(16,76),(17,83),(18,78),(19,73),(20,80),(21,75),(22,82),(23,77),(24,84),(25,95),(26,90),(27,85),(28,92),(29,87),(30,94),(31,89),(32,96),(33,91),(34,86),(35,93),(36,88),(37,50),(38,57),(39,52),(40,59),(41,54),(42,49),(43,56),(44,51),(45,58),(46,53),(47,60),(48,55)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×D4 | D4⋊2S3 | D4⋊6D6 |
| kernel | Dic6⋊11D4 | C4×Dic6 | C4×D12 | D4×Dic3 | D6⋊3D4 | C23.14D6 | C12⋊3D4 | C3×C4⋊1D4 | C2×D4⋊2S3 | C4⋊1D4 | Dic6 | C42 | C2×D4 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 2 | 1 | 4 | 1 | 6 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_{11}D_4 % in TeX
G:=Group("Dic6:11D4"); // GroupNames label
G:=SmallGroup(192,1277);
// by ID
G=gap.SmallGroup(192,1277);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations