metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊3Q8, C42.33D6, C12.13SD16, C4⋊C8⋊9S3, C4.43(S3×Q8), C3⋊2(D4⋊2Q8), C8⋊Dic3⋊16C2, (C2×C8).129D6, C12⋊2Q8⋊12C2, (C4×D12).11C2, (C2×C12).121D4, (C2×C4).132D12, C12.102(C2×Q8), C6.11(C2×SD16), C2.D24.5C2, C4.13(C24⋊C2), C2.16(C8⋊D6), C6.13(C8⋊C22), (C4×C12).68C22, C6.30(C22⋊Q8), C12.286(C4○D4), (C2×C12).752C23, (C2×C24).139C22, C2.11(C4.D12), C22.115(C2×D12), C4⋊Dic3.18C22, C4.110(D4⋊2S3), (C2×D12).195C22, (C3×C4⋊C8)⋊11C2, C2.14(C2×C24⋊C2), (C2×C6).135(C2×D4), (C2×C4).697(C22×S3), SmallGroup(192,401)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊3Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a7b, dbd-1=a6b, dcd-1=c-1 >
Subgroups: 360 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, C2×C8, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, D4⋊2Q8, C8⋊Dic3, C2.D24, C3×C4⋊C8, C12⋊2Q8, C4×D12, D12⋊3Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×SD16, C8⋊C22, C24⋊C2, C2×D12, D4⋊2S3, S3×Q8, D4⋊2Q8, C4.D12, C2×C24⋊C2, C8⋊D6, D12⋊3Q8
►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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 96)(45 95)(46 94)(47 93)(48 92)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 84)(58 83)(59 82)(60 81)
(1 84 64 55)(2 83 65 54)(3 82 66 53)(4 81 67 52)(5 80 68 51)(6 79 69 50)(7 78 70 49)(8 77 71 60)(9 76 72 59)(10 75 61 58)(11 74 62 57)(12 73 63 56)(13 37 28 95)(14 48 29 94)(15 47 30 93)(16 46 31 92)(17 45 32 91)(18 44 33 90)(19 43 34 89)(20 42 35 88)(21 41 36 87)(22 40 25 86)(23 39 26 85)(24 38 27 96)
(1 28 64 13)(2 29 65 14)(3 30 66 15)(4 31 67 16)(5 32 68 17)(6 33 69 18)(7 34 70 19)(8 35 71 20)(9 36 72 21)(10 25 61 22)(11 26 62 23)(12 27 63 24)(37 55 95 84)(38 56 96 73)(39 57 85 74)(40 58 86 75)(41 59 87 76)(42 60 88 77)(43 49 89 78)(44 50 90 79)(45 51 91 80)(46 52 92 81)(47 53 93 82)(48 54 94 83)
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,96)(45,95)(46,94)(47,93)(48,92)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,84)(58,83)(59,82)(60,81), (1,84,64,55)(2,83,65,54)(3,82,66,53)(4,81,67,52)(5,80,68,51)(6,79,69,50)(7,78,70,49)(8,77,71,60)(9,76,72,59)(10,75,61,58)(11,74,62,57)(12,73,63,56)(13,37,28,95)(14,48,29,94)(15,47,30,93)(16,46,31,92)(17,45,32,91)(18,44,33,90)(19,43,34,89)(20,42,35,88)(21,41,36,87)(22,40,25,86)(23,39,26,85)(24,38,27,96), (1,28,64,13)(2,29,65,14)(3,30,66,15)(4,31,67,16)(5,32,68,17)(6,33,69,18)(7,34,70,19)(8,35,71,20)(9,36,72,21)(10,25,61,22)(11,26,62,23)(12,27,63,24)(37,55,95,84)(38,56,96,73)(39,57,85,74)(40,58,86,75)(41,59,87,76)(42,60,88,77)(43,49,89,78)(44,50,90,79)(45,51,91,80)(46,52,92,81)(47,53,93,82)(48,54,94,83)>;
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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(37,91)(38,90)(39,89)(40,88)(41,87)(42,86)(43,85)(44,96)(45,95)(46,94)(47,93)(48,92)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,84)(58,83)(59,82)(60,81), (1,84,64,55)(2,83,65,54)(3,82,66,53)(4,81,67,52)(5,80,68,51)(6,79,69,50)(7,78,70,49)(8,77,71,60)(9,76,72,59)(10,75,61,58)(11,74,62,57)(12,73,63,56)(13,37,28,95)(14,48,29,94)(15,47,30,93)(16,46,31,92)(17,45,32,91)(18,44,33,90)(19,43,34,89)(20,42,35,88)(21,41,36,87)(22,40,25,86)(23,39,26,85)(24,38,27,96), (1,28,64,13)(2,29,65,14)(3,30,66,15)(4,31,67,16)(5,32,68,17)(6,33,69,18)(7,34,70,19)(8,35,71,20)(9,36,72,21)(10,25,61,22)(11,26,62,23)(12,27,63,24)(37,55,95,84)(38,56,96,73)(39,57,85,74)(40,58,86,75)(41,59,87,76)(42,60,88,77)(43,49,89,78)(44,50,90,79)(45,51,91,80)(46,52,92,81)(47,53,93,82)(48,54,94,83) );
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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,72),(11,71),(12,70),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(37,91),(38,90),(39,89),(40,88),(41,87),(42,86),(43,85),(44,96),(45,95),(46,94),(47,93),(48,92),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,84),(58,83),(59,82),(60,81)], [(1,84,64,55),(2,83,65,54),(3,82,66,53),(4,81,67,52),(5,80,68,51),(6,79,69,50),(7,78,70,49),(8,77,71,60),(9,76,72,59),(10,75,61,58),(11,74,62,57),(12,73,63,56),(13,37,28,95),(14,48,29,94),(15,47,30,93),(16,46,31,92),(17,45,32,91),(18,44,33,90),(19,43,34,89),(20,42,35,88),(21,41,36,87),(22,40,25,86),(23,39,26,85),(24,38,27,96)], [(1,28,64,13),(2,29,65,14),(3,30,66,15),(4,31,67,16),(5,32,68,17),(6,33,69,18),(7,34,70,19),(8,35,71,20),(9,36,72,21),(10,25,61,22),(11,26,62,23),(12,27,63,24),(37,55,95,84),(38,56,96,73),(39,57,85,74),(40,58,86,75),(41,59,87,76),(42,60,88,77),(43,49,89,78),(44,50,90,79),(45,51,91,80),(46,52,92,81),(47,53,93,82),(48,54,94,83)]])
39 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 | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| 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 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 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 | Q8 | D4 | D6 | D6 | SD16 | C4○D4 | D12 | C24⋊C2 | C8⋊C22 | D4⋊2S3 | S3×Q8 | C8⋊D6 |
| kernel | D12⋊3Q8 | C8⋊Dic3 | C2.D24 | C3×C4⋊C8 | C12⋊2Q8 | C4×D12 | C4⋊C8 | D12 | C2×C12 | C42 | C2×C8 | C12 | C12 | C2×C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of D12⋊3Q8 ►in GL4(𝔽73) generated by
| 7 | 66 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 36 | 11 | 0 | 0 |
| 48 | 37 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [7,7,0,0,66,14,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,1,72,0,0,0,0,1,0,0,0,0,72],[36,48,0,0,11,37,0,0,0,0,0,72,0,0,1,0],[66,14,0,0,59,7,0,0,0,0,27,0,0,0,0,46] >;
D12⋊3Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_3Q_8 % in TeX
G:=Group("D12:3Q8"); // GroupNames label
G:=SmallGroup(192,401);
// by ID
G=gap.SmallGroup(192,401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations