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, C4⋊1(D4⋊2S3), C12⋊3D4⋊27C2, (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, C23.78(C22×S3), (C22×C6).76C23, (C2×D12).270C22, C4⋊Dic3.381C22, C22.283(S3×C23), Dic3⋊C4.164C22, (C22×S3).116C23, (C2×Dic6).301C22, (C4×Dic3).155C22, (C2×Dic3).269C23, 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
Generators and relations for Dic6⋊11D4
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 >
Subgroups: 832 in 312 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C4⋊1D4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, Q8⋊6D4, C4×Dic6, C4×D12, D4×Dic3, D6⋊3D4, C23.14D6, C12⋊3D4, C3×C4⋊1D4, C2×D4⋊2S3, Dic6⋊11D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, S3×D4, D4⋊2S3, S3×C23, Q8⋊6D4, C2×S3×D4, C2×D4⋊2S3, D4⋊6D6, Dic6⋊11D4
►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 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 72 12 66)(13 73 19 79)(14 84 20 78)(15 83 21 77)(16 82 22 76)(17 81 23 75)(18 80 24 74)(25 42 31 48)(26 41 32 47)(27 40 33 46)(28 39 34 45)(29 38 35 44)(30 37 36 43)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)
(1 14 46 60)(2 15 47 49)(3 16 48 50)(4 17 37 51)(5 18 38 52)(6 19 39 53)(7 20 40 54)(8 21 41 55)(9 22 42 56)(10 23 43 57)(11 24 44 58)(12 13 45 59)(25 89 63 82)(26 90 64 83)(27 91 65 84)(28 92 66 73)(29 93 67 74)(30 94 68 75)(31 95 69 76)(32 96 70 77)(33 85 71 78)(34 86 72 79)(35 87 61 80)(36 88 62 81)
(1 91)(2 86)(3 93)(4 88)(5 95)(6 90)(7 85)(8 92)(9 87)(10 94)(11 89)(12 96)(13 32)(14 27)(15 34)(16 29)(17 36)(18 31)(19 26)(20 33)(21 28)(22 35)(23 30)(24 25)(37 81)(38 76)(39 83)(40 78)(41 73)(42 80)(43 75)(44 82)(45 77)(46 84)(47 79)(48 74)(49 72)(50 67)(51 62)(52 69)(53 64)(54 71)(55 66)(56 61)(57 68)(58 63)(59 70)(60 65)
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), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,73,19,79)(14,84,20,78)(15,83,21,77)(16,82,22,76)(17,81,23,75)(18,80,24,74)(25,42,31,48)(26,41,32,47)(27,40,33,46)(28,39,34,45)(29,38,35,44)(30,37,36,43)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91), (1,14,46,60)(2,15,47,49)(3,16,48,50)(4,17,37,51)(5,18,38,52)(6,19,39,53)(7,20,40,54)(8,21,41,55)(9,22,42,56)(10,23,43,57)(11,24,44,58)(12,13,45,59)(25,89,63,82)(26,90,64,83)(27,91,65,84)(28,92,66,73)(29,93,67,74)(30,94,68,75)(31,95,69,76)(32,96,70,77)(33,85,71,78)(34,86,72,79)(35,87,61,80)(36,88,62,81), (1,91)(2,86)(3,93)(4,88)(5,95)(6,90)(7,85)(8,92)(9,87)(10,94)(11,89)(12,96)(13,32)(14,27)(15,34)(16,29)(17,36)(18,31)(19,26)(20,33)(21,28)(22,35)(23,30)(24,25)(37,81)(38,76)(39,83)(40,78)(41,73)(42,80)(43,75)(44,82)(45,77)(46,84)(47,79)(48,74)(49,72)(50,67)(51,62)(52,69)(53,64)(54,71)(55,66)(56,61)(57,68)(58,63)(59,70)(60,65)>;
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), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,73,19,79)(14,84,20,78)(15,83,21,77)(16,82,22,76)(17,81,23,75)(18,80,24,74)(25,42,31,48)(26,41,32,47)(27,40,33,46)(28,39,34,45)(29,38,35,44)(30,37,36,43)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91), (1,14,46,60)(2,15,47,49)(3,16,48,50)(4,17,37,51)(5,18,38,52)(6,19,39,53)(7,20,40,54)(8,21,41,55)(9,22,42,56)(10,23,43,57)(11,24,44,58)(12,13,45,59)(25,89,63,82)(26,90,64,83)(27,91,65,84)(28,92,66,73)(29,93,67,74)(30,94,68,75)(31,95,69,76)(32,96,70,77)(33,85,71,78)(34,86,72,79)(35,87,61,80)(36,88,62,81), (1,91)(2,86)(3,93)(4,88)(5,95)(6,90)(7,85)(8,92)(9,87)(10,94)(11,89)(12,96)(13,32)(14,27)(15,34)(16,29)(17,36)(18,31)(19,26)(20,33)(21,28)(22,35)(23,30)(24,25)(37,81)(38,76)(39,83)(40,78)(41,73)(42,80)(43,75)(44,82)(45,77)(46,84)(47,79)(48,74)(49,72)(50,67)(51,62)(52,69)(53,64)(54,71)(55,66)(56,61)(57,68)(58,63)(59,70)(60,65) );
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)], [(1,65,7,71),(2,64,8,70),(3,63,9,69),(4,62,10,68),(5,61,11,67),(6,72,12,66),(13,73,19,79),(14,84,20,78),(15,83,21,77),(16,82,22,76),(17,81,23,75),(18,80,24,74),(25,42,31,48),(26,41,32,47),(27,40,33,46),(28,39,34,45),(29,38,35,44),(30,37,36,43),(49,90,55,96),(50,89,56,95),(51,88,57,94),(52,87,58,93),(53,86,59,92),(54,85,60,91)], [(1,14,46,60),(2,15,47,49),(3,16,48,50),(4,17,37,51),(5,18,38,52),(6,19,39,53),(7,20,40,54),(8,21,41,55),(9,22,42,56),(10,23,43,57),(11,24,44,58),(12,13,45,59),(25,89,63,82),(26,90,64,83),(27,91,65,84),(28,92,66,73),(29,93,67,74),(30,94,68,75),(31,95,69,76),(32,96,70,77),(33,85,71,78),(34,86,72,79),(35,87,61,80),(36,88,62,81)], [(1,91),(2,86),(3,93),(4,88),(5,95),(6,90),(7,85),(8,92),(9,87),(10,94),(11,89),(12,96),(13,32),(14,27),(15,34),(16,29),(17,36),(18,31),(19,26),(20,33),(21,28),(22,35),(23,30),(24,25),(37,81),(38,76),(39,83),(40,78),(41,73),(42,80),(43,75),(44,82),(45,77),(46,84),(47,79),(48,74),(49,72),(50,67),(51,62),(52,69),(53,64),(54,71),(55,66),(56,61),(57,68),(58,63),(59,70),(60,65)]])
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 |
Matrix representation of Dic6⋊11D4 ►in 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] >;
Dic6⋊11D4 in GAP, Magma, Sage, TeX
{\rm 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