metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.4Q8, C42.69D6, C4⋊C4.75D6, C3⋊7(D4.Q8), C4.10(S3×Q8), C12⋊C8⋊30C2, C12.34(C2×Q8), C42.C2⋊1S3, (C4×D12).16C2, (C2×C12).275D4, C6.Q16⋊42C2, C12.70(C4○D4), C6.109(C4○D8), C12.Q8⋊41C2, C6.D8.12C2, C6.74(C22⋊Q8), C2.21(D4⋊D6), C6.122(C8⋊C22), (C2×C12).384C23, (C4×C12).114C22, C2.11(D6⋊3Q8), C4.33(Q8⋊3S3), C2.28(Q8.13D6), (C2×D12).245C22, C4⋊Dic3.343C22, (C2×C6).515(C2×D4), (C3×C42.C2)⋊1C2, (C2×C4).66(C3⋊D4), (C2×C3⋊C8).126C22, (C3×C4⋊C4).122C22, (C2×C4).482(C22×S3), C22.188(C2×C3⋊D4), SmallGroup(192,625)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.4Q8
G = < a,b,c,d | a12=b2=c4=1, d2=a6c2, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 304 in 102 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C3⋊C8, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, D4.Q8, C12⋊C8, C6.Q16, C12.Q8, C6.D8, C4×D12, C3×C42.C2, D12.4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C3⋊D4, C22×S3, C22⋊Q8, C4○D8, C8⋊C22, S3×Q8, Q8⋊3S3, C2×C3⋊D4, D4.Q8, D6⋊3Q8, D4⋊D6, Q8.13D6, D12.4Q8
►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 95)(2 94)(3 93)(4 92)(5 91)(6 90)(7 89)(8 88)(9 87)(10 86)(11 85)(12 96)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(49 75)(50 74)(51 73)(52 84)(53 83)(54 82)(55 81)(56 80)(57 79)(58 78)(59 77)(60 76)
(1 17 90 46)(2 24 91 41)(3 19 92 48)(4 14 93 43)(5 21 94 38)(6 16 95 45)(7 23 96 40)(8 18 85 47)(9 13 86 42)(10 20 87 37)(11 15 88 44)(12 22 89 39)(25 56 72 77)(26 51 61 84)(27 58 62 79)(28 53 63 74)(29 60 64 81)(30 55 65 76)(31 50 66 83)(32 57 67 78)(33 52 68 73)(34 59 69 80)(35 54 70 75)(36 49 71 82)
(1 82 96 55)(2 83 85 56)(3 84 86 57)(4 73 87 58)(5 74 88 59)(6 75 89 60)(7 76 90 49)(8 77 91 50)(9 78 92 51)(10 79 93 52)(11 80 94 53)(12 81 95 54)(13 67 48 26)(14 68 37 27)(15 69 38 28)(16 70 39 29)(17 71 40 30)(18 72 41 31)(19 61 42 32)(20 62 43 33)(21 63 44 34)(22 64 45 35)(23 65 46 36)(24 66 47 25)
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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,96)(13,40)(14,39)(15,38)(16,37)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(49,75)(50,74)(51,73)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76), (1,17,90,46)(2,24,91,41)(3,19,92,48)(4,14,93,43)(5,21,94,38)(6,16,95,45)(7,23,96,40)(8,18,85,47)(9,13,86,42)(10,20,87,37)(11,15,88,44)(12,22,89,39)(25,56,72,77)(26,51,61,84)(27,58,62,79)(28,53,63,74)(29,60,64,81)(30,55,65,76)(31,50,66,83)(32,57,67,78)(33,52,68,73)(34,59,69,80)(35,54,70,75)(36,49,71,82), (1,82,96,55)(2,83,85,56)(3,84,86,57)(4,73,87,58)(5,74,88,59)(6,75,89,60)(7,76,90,49)(8,77,91,50)(9,78,92,51)(10,79,93,52)(11,80,94,53)(12,81,95,54)(13,67,48,26)(14,68,37,27)(15,69,38,28)(16,70,39,29)(17,71,40,30)(18,72,41,31)(19,61,42,32)(20,62,43,33)(21,63,44,34)(22,64,45,35)(23,65,46,36)(24,66,47,25)>;
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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,90)(7,89)(8,88)(9,87)(10,86)(11,85)(12,96)(13,40)(14,39)(15,38)(16,37)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(49,75)(50,74)(51,73)(52,84)(53,83)(54,82)(55,81)(56,80)(57,79)(58,78)(59,77)(60,76), (1,17,90,46)(2,24,91,41)(3,19,92,48)(4,14,93,43)(5,21,94,38)(6,16,95,45)(7,23,96,40)(8,18,85,47)(9,13,86,42)(10,20,87,37)(11,15,88,44)(12,22,89,39)(25,56,72,77)(26,51,61,84)(27,58,62,79)(28,53,63,74)(29,60,64,81)(30,55,65,76)(31,50,66,83)(32,57,67,78)(33,52,68,73)(34,59,69,80)(35,54,70,75)(36,49,71,82), (1,82,96,55)(2,83,85,56)(3,84,86,57)(4,73,87,58)(5,74,88,59)(6,75,89,60)(7,76,90,49)(8,77,91,50)(9,78,92,51)(10,79,93,52)(11,80,94,53)(12,81,95,54)(13,67,48,26)(14,68,37,27)(15,69,38,28)(16,70,39,29)(17,71,40,30)(18,72,41,31)(19,61,42,32)(20,62,43,33)(21,63,44,34)(22,64,45,35)(23,65,46,36)(24,66,47,25) );
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)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,90),(7,89),(8,88),(9,87),(10,86),(11,85),(12,96),(13,40),(14,39),(15,38),(16,37),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,42),(24,41),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(49,75),(50,74),(51,73),(52,84),(53,83),(54,82),(55,81),(56,80),(57,79),(58,78),(59,77),(60,76)], [(1,17,90,46),(2,24,91,41),(3,19,92,48),(4,14,93,43),(5,21,94,38),(6,16,95,45),(7,23,96,40),(8,18,85,47),(9,13,86,42),(10,20,87,37),(11,15,88,44),(12,22,89,39),(25,56,72,77),(26,51,61,84),(27,58,62,79),(28,53,63,74),(29,60,64,81),(30,55,65,76),(31,50,66,83),(32,57,67,78),(33,52,68,73),(34,59,69,80),(35,54,70,75),(36,49,71,82)], [(1,82,96,55),(2,83,85,56),(3,84,86,57),(4,73,87,58),(5,74,88,59),(6,75,89,60),(7,76,90,49),(8,77,91,50),(9,78,92,51),(10,79,93,52),(11,80,94,53),(12,81,95,54),(13,67,48,26),(14,68,37,27),(15,69,38,28),(16,70,39,29),(17,71,40,30),(18,72,41,31),(19,61,42,32),(20,62,43,33),(21,63,44,34),(22,64,45,35),(23,65,46,36),(24,66,47,25)]])
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 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | C8⋊C22 | S3×Q8 | Q8⋊3S3 | D4⋊D6 | Q8.13D6 |
| kernel | D12.4Q8 | C12⋊C8 | C6.Q16 | C12.Q8 | C6.D8 | C4×D12 | C3×C42.C2 | C42.C2 | D12 | C2×C12 | C42 | C4⋊C4 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.4Q8 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 24 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 37 | 3 | 0 | 0 | 0 | 0 |
| 30 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 64 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[1,24,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,2,72],[37,30,0,0,0,0,3,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,32,0],[27,64,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46] >;
D12.4Q8 in GAP, Magma, Sage, TeX
D_{12}._4Q_8 % in TeX
G:=Group("D12.4Q8"); // GroupNames label
G:=SmallGroup(192,625);
// by ID
G=gap.SmallGroup(192,625);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,344,254,219,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6*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