metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊6Q8, C12.18D8, C42.81D6, C4⋊Q8⋊5S3, C4⋊C4.83D6, C6.60(C2×D8), C4.12(S3×Q8), C3⋊5(D4⋊Q8), C12⋊C8⋊35C2, C12.39(C2×Q8), C4.16(D4⋊S3), (C4×D12).19C2, C6.Q16⋊43C2, (C2×C12).157D4, C12.82(C4○D4), C6.D8.15C2, C6.76(C22⋊Q8), (C4×C12).134C22, (C2×C12).405C23, C2.13(D6⋊3Q8), C4.35(Q8⋊3S3), C6.97(C8.C22), (C2×D12).249C22, C4⋊Dic3.348C22, C2.18(Q8.11D6), (C3×C4⋊Q8)⋊5C2, C2.15(C2×D4⋊S3), (C2×C6).536(C2×D4), (C2×C3⋊C8).138C22, (C2×C4).189(C3⋊D4), (C3×C4⋊C4).130C22, (C2×C4).502(C22×S3), C22.208(C2×C3⋊D4), SmallGroup(192,646)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊6Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 320 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, D4⋊Q8, C12⋊C8, C6.Q16, C6.D8, C4×D12, C3×C4⋊Q8, D12⋊6Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C2×D8, C8.C22, D4⋊S3, S3×Q8, Q8⋊3S3, C2×C3⋊D4, D4⋊Q8, C2×D4⋊S3, Q8.11D6, D6⋊3Q8, D12⋊6Q8
►On 96 points
Generators in S96
(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 27)(2 26)(3 25)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 84)(24 83)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 96)(48 95)(49 62)(50 61)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)
(1 66 28 49)(2 61 29 56)(3 68 30 51)(4 63 31 58)(5 70 32 53)(6 65 33 60)(7 72 34 55)(8 67 35 50)(9 62 36 57)(10 69 25 52)(11 64 26 59)(12 71 27 54)(13 39 83 96)(14 46 84 91)(15 41 73 86)(16 48 74 93)(17 43 75 88)(18 38 76 95)(19 45 77 90)(20 40 78 85)(21 47 79 92)(22 42 80 87)(23 37 81 94)(24 44 82 89)
(1 13 28 83)(2 14 29 84)(3 15 30 73)(4 16 31 74)(5 17 32 75)(6 18 33 76)(7 19 34 77)(8 20 35 78)(9 21 36 79)(10 22 25 80)(11 23 26 81)(12 24 27 82)(37 64 94 59)(38 65 95 60)(39 66 96 49)(40 67 85 50)(41 68 86 51)(42 69 87 52)(43 70 88 53)(44 71 89 54)(45 72 90 55)(46 61 91 56)(47 62 92 57)(48 63 93 58)
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,96)(48,95)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63), (1,66,28,49)(2,61,29,56)(3,68,30,51)(4,63,31,58)(5,70,32,53)(6,65,33,60)(7,72,34,55)(8,67,35,50)(9,62,36,57)(10,69,25,52)(11,64,26,59)(12,71,27,54)(13,39,83,96)(14,46,84,91)(15,41,73,86)(16,48,74,93)(17,43,75,88)(18,38,76,95)(19,45,77,90)(20,40,78,85)(21,47,79,92)(22,42,80,87)(23,37,81,94)(24,44,82,89), (1,13,28,83)(2,14,29,84)(3,15,30,73)(4,16,31,74)(5,17,32,75)(6,18,33,76)(7,19,34,77)(8,20,35,78)(9,21,36,79)(10,22,25,80)(11,23,26,81)(12,24,27,82)(37,64,94,59)(38,65,95,60)(39,66,96,49)(40,67,85,50)(41,68,86,51)(42,69,87,52)(43,70,88,53)(44,71,89,54)(45,72,90,55)(46,61,91,56)(47,62,92,57)(48,63,93,58)>;
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), (1,27)(2,26)(3,25)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,84)(24,83)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,96)(48,95)(49,62)(50,61)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63), (1,66,28,49)(2,61,29,56)(3,68,30,51)(4,63,31,58)(5,70,32,53)(6,65,33,60)(7,72,34,55)(8,67,35,50)(9,62,36,57)(10,69,25,52)(11,64,26,59)(12,71,27,54)(13,39,83,96)(14,46,84,91)(15,41,73,86)(16,48,74,93)(17,43,75,88)(18,38,76,95)(19,45,77,90)(20,40,78,85)(21,47,79,92)(22,42,80,87)(23,37,81,94)(24,44,82,89), (1,13,28,83)(2,14,29,84)(3,15,30,73)(4,16,31,74)(5,17,32,75)(6,18,33,76)(7,19,34,77)(8,20,35,78)(9,21,36,79)(10,22,25,80)(11,23,26,81)(12,24,27,82)(37,64,94,59)(38,65,95,60)(39,66,96,49)(40,67,85,50)(41,68,86,51)(42,69,87,52)(43,70,88,53)(44,71,89,54)(45,72,90,55)(46,61,91,56)(47,62,92,57)(48,63,93,58) );
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)], [(1,27),(2,26),(3,25),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,82),(14,81),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,84),(24,83),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,96),(48,95),(49,62),(50,61),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63)], [(1,66,28,49),(2,61,29,56),(3,68,30,51),(4,63,31,58),(5,70,32,53),(6,65,33,60),(7,72,34,55),(8,67,35,50),(9,62,36,57),(10,69,25,52),(11,64,26,59),(12,71,27,54),(13,39,83,96),(14,46,84,91),(15,41,73,86),(16,48,74,93),(17,43,75,88),(18,38,76,95),(19,45,77,90),(20,40,78,85),(21,47,79,92),(22,42,80,87),(23,37,81,94),(24,44,82,89)], [(1,13,28,83),(2,14,29,84),(3,15,30,73),(4,16,31,74),(5,17,32,75),(6,18,33,76),(7,19,34,77),(8,20,35,78),(9,21,36,79),(10,22,25,80),(11,23,26,81),(12,24,27,82),(37,64,94,59),(38,65,95,60),(39,66,96,49),(40,67,85,50),(41,68,86,51),(42,69,87,52),(43,70,88,53),(44,71,89,54),(45,72,90,55),(46,61,91,56),(47,62,92,57),(48,63,93,58)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 12 | 12 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | D8 | C4○D4 | C3⋊D4 | C8.C22 | D4⋊S3 | S3×Q8 | Q8⋊3S3 | Q8.11D6 |
| kernel | D12⋊6Q8 | C12⋊C8 | C6.Q16 | C6.D8 | C4×D12 | C3×C4⋊Q8 | C4⋊Q8 | D12 | C2×C12 | C42 | C4⋊C4 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of D12⋊6Q8 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 3 |
| 0 | 0 | 0 | 0 | 48 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 59 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 3 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 21 | 3 | 0 | 0 | 0 | 0 |
| 23 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 13 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 25 |
| 0 | 0 | 0 | 0 | 38 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 60 | 46 | 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 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,48,0,0,0,0,3,1],[1,59,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,3,1],[21,23,0,0,0,0,3,52,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,0,0,0,0,0,38,0,0,0,0,25,0],[27,60,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D12⋊6Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_6Q_8 % in TeX
G:=Group("D12:6Q8"); // GroupNames label
G:=SmallGroup(192,646);
// by ID
G=gap.SmallGroup(192,646);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations