metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.7D12, Dic6.9D4, D6⋊C8⋊6C2, C4⋊C4.14D6, (C3×D4).2D4, (C2×C8).10D6, C4.4(C2×D12), C12.3(C2×D4), C4.87(S3×D4), D4⋊C4⋊6S3, C4.D12⋊4C2, C6.22C22≀C2, (C2×Dic12)⋊5C2, (C2×D4).137D6, C6.24(C4○D8), C6.SD16⋊7C2, C3⋊2(D4.7D4), (C2×C24).10C22, (C22×S3).15D4, (C6×D4).44C22, C22.181(S3×D4), C2.25(D6⋊D4), C2.10(D8⋊3S3), (C2×C12).223C23, (C2×Dic3).143D4, C2.13(D4.D6), C6.31(C8.C22), (C2×Dic6).60C22, (C2×D4.S3)⋊6C2, (C3×D4⋊C4)⋊6C2, (C2×C6).236(C2×D4), (C2×C3⋊C8).21C22, (S3×C2×C4).15C22, (C2×D4⋊2S3).5C2, (C3×C4⋊C4).24C22, (C2×C4).330(C22×S3), SmallGroup(192,342)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D12
G = < a,b,c,d | a4=b2=c12=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 456 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, Dic12, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4.S3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, D4.7D4, C6.SD16, D6⋊C8, C3×D4⋊C4, C4.D12, C2×Dic12, C2×D4.S3, C2×D4⋊2S3, D4.D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C4○D8, C8.C22, C2×D12, S3×D4, D4.7D4, D6⋊D4, D8⋊3S3, D4.D6, D4.D12
►On 96 points
(1 67 45 13)(2 14 46 68)(3 69 47 15)(4 16 48 70)(5 71 37 17)(6 18 38 72)(7 61 39 19)(8 20 40 62)(9 63 41 21)(10 22 42 64)(11 65 43 23)(12 24 44 66)(25 80 93 51)(26 52 94 81)(27 82 95 53)(28 54 96 83)(29 84 85 55)(30 56 86 73)(31 74 87 57)(32 58 88 75)(33 76 89 59)(34 60 90 77)(35 78 91 49)(36 50 92 79)
(1 92)(2 80)(3 94)(4 82)(5 96)(6 84)(7 86)(8 74)(9 88)(10 76)(11 90)(12 78)(13 79)(14 25)(15 81)(16 27)(17 83)(18 29)(19 73)(20 31)(21 75)(22 33)(23 77)(24 35)(26 47)(28 37)(30 39)(32 41)(34 43)(36 45)(38 55)(40 57)(42 59)(44 49)(46 51)(48 53)(50 67)(52 69)(54 71)(56 61)(58 63)(60 65)(62 87)(64 89)(66 91)(68 93)(70 95)(72 85)
(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 44 45 12)(2 11 46 43)(3 42 47 10)(4 9 48 41)(5 40 37 8)(6 7 38 39)(13 66 67 24)(14 23 68 65)(15 64 69 22)(16 21 70 63)(17 62 71 20)(18 19 72 61)(25 90 93 34)(26 33 94 89)(27 88 95 32)(28 31 96 87)(29 86 85 30)(35 92 91 36)(49 79 78 50)(51 77 80 60)(52 59 81 76)(53 75 82 58)(54 57 83 74)(55 73 84 56)
G:=sub<Sym(96)| (1,67,45,13)(2,14,46,68)(3,69,47,15)(4,16,48,70)(5,71,37,17)(6,18,38,72)(7,61,39,19)(8,20,40,62)(9,63,41,21)(10,22,42,64)(11,65,43,23)(12,24,44,66)(25,80,93,51)(26,52,94,81)(27,82,95,53)(28,54,96,83)(29,84,85,55)(30,56,86,73)(31,74,87,57)(32,58,88,75)(33,76,89,59)(34,60,90,77)(35,78,91,49)(36,50,92,79), (1,92)(2,80)(3,94)(4,82)(5,96)(6,84)(7,86)(8,74)(9,88)(10,76)(11,90)(12,78)(13,79)(14,25)(15,81)(16,27)(17,83)(18,29)(19,73)(20,31)(21,75)(22,33)(23,77)(24,35)(26,47)(28,37)(30,39)(32,41)(34,43)(36,45)(38,55)(40,57)(42,59)(44,49)(46,51)(48,53)(50,67)(52,69)(54,71)(56,61)(58,63)(60,65)(62,87)(64,89)(66,91)(68,93)(70,95)(72,85), (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), (1,44,45,12)(2,11,46,43)(3,42,47,10)(4,9,48,41)(5,40,37,8)(6,7,38,39)(13,66,67,24)(14,23,68,65)(15,64,69,22)(16,21,70,63)(17,62,71,20)(18,19,72,61)(25,90,93,34)(26,33,94,89)(27,88,95,32)(28,31,96,87)(29,86,85,30)(35,92,91,36)(49,79,78,50)(51,77,80,60)(52,59,81,76)(53,75,82,58)(54,57,83,74)(55,73,84,56)>;
G:=Group( (1,67,45,13)(2,14,46,68)(3,69,47,15)(4,16,48,70)(5,71,37,17)(6,18,38,72)(7,61,39,19)(8,20,40,62)(9,63,41,21)(10,22,42,64)(11,65,43,23)(12,24,44,66)(25,80,93,51)(26,52,94,81)(27,82,95,53)(28,54,96,83)(29,84,85,55)(30,56,86,73)(31,74,87,57)(32,58,88,75)(33,76,89,59)(34,60,90,77)(35,78,91,49)(36,50,92,79), (1,92)(2,80)(3,94)(4,82)(5,96)(6,84)(7,86)(8,74)(9,88)(10,76)(11,90)(12,78)(13,79)(14,25)(15,81)(16,27)(17,83)(18,29)(19,73)(20,31)(21,75)(22,33)(23,77)(24,35)(26,47)(28,37)(30,39)(32,41)(34,43)(36,45)(38,55)(40,57)(42,59)(44,49)(46,51)(48,53)(50,67)(52,69)(54,71)(56,61)(58,63)(60,65)(62,87)(64,89)(66,91)(68,93)(70,95)(72,85), (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), (1,44,45,12)(2,11,46,43)(3,42,47,10)(4,9,48,41)(5,40,37,8)(6,7,38,39)(13,66,67,24)(14,23,68,65)(15,64,69,22)(16,21,70,63)(17,62,71,20)(18,19,72,61)(25,90,93,34)(26,33,94,89)(27,88,95,32)(28,31,96,87)(29,86,85,30)(35,92,91,36)(49,79,78,50)(51,77,80,60)(52,59,81,76)(53,75,82,58)(54,57,83,74)(55,73,84,56) );
G=PermutationGroup([[(1,67,45,13),(2,14,46,68),(3,69,47,15),(4,16,48,70),(5,71,37,17),(6,18,38,72),(7,61,39,19),(8,20,40,62),(9,63,41,21),(10,22,42,64),(11,65,43,23),(12,24,44,66),(25,80,93,51),(26,52,94,81),(27,82,95,53),(28,54,96,83),(29,84,85,55),(30,56,86,73),(31,74,87,57),(32,58,88,75),(33,76,89,59),(34,60,90,77),(35,78,91,49),(36,50,92,79)], [(1,92),(2,80),(3,94),(4,82),(5,96),(6,84),(7,86),(8,74),(9,88),(10,76),(11,90),(12,78),(13,79),(14,25),(15,81),(16,27),(17,83),(18,29),(19,73),(20,31),(21,75),(22,33),(23,77),(24,35),(26,47),(28,37),(30,39),(32,41),(34,43),(36,45),(38,55),(40,57),(42,59),(44,49),(46,51),(48,53),(50,67),(52,69),(54,71),(56,61),(58,63),(60,65),(62,87),(64,89),(66,91),(68,93),(70,95),(72,85)], [(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)], [(1,44,45,12),(2,11,46,43),(3,42,47,10),(4,9,48,41),(5,40,37,8),(6,7,38,39),(13,66,67,24),(14,23,68,65),(15,64,69,22),(16,21,70,63),(17,62,71,20),(18,19,72,61),(25,90,93,34),(26,33,94,89),(27,88,95,32),(28,31,96,87),(29,86,85,30),(35,92,91,36),(49,79,78,50),(51,77,80,60),(52,59,81,76),(53,75,82,58),(54,57,83,74),(55,73,84,56)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 2 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | D12 | C4○D8 | C8.C22 | S3×D4 | S3×D4 | D8⋊3S3 | D4.D6 |
| kernel | D4.D12 | C6.SD16 | D6⋊C8 | C3×D4⋊C4 | C4.D12 | C2×Dic12 | C2×D4.S3 | C2×D4⋊2S3 | D4⋊C4 | Dic6 | C2×Dic3 | C3×D4 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D4.D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 65 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 34 |
| 0 | 0 | 18 | 22 |
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 72 | 48 |
| 0 | 0 | 0 | 1 |
| 7 | 66 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 72 | 48 |
| 0 | 0 | 3 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,27,65,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,51,18,0,0,34,22],[66,66,0,0,7,59,0,0,0,0,72,0,0,0,48,1],[7,59,0,0,66,66,0,0,0,0,72,3,0,0,48,1] >;
D4.D12 in GAP, Magma, Sage, TeX
D_4.D_{12} % in TeX
G:=Group("D4.D12"); // GroupNames label
G:=SmallGroup(192,342);
// by ID
G=gap.SmallGroup(192,342);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,254,219,226,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations