direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×Dic6, C42.101D6, C6.552- (1+4), C3⋊1(D4×Q8), (C3×D4)⋊5Q8, C12⋊1(C2×Q8), C12⋊Q8⋊14C2, C4⋊1(C2×Dic6), C4⋊C4.276D6, (C4×D4).10S3, C4.138(S3×D4), C12⋊2Q8⋊21C2, (C4×Dic6)⋊24C2, (D4×C12).11C2, (C2×D4).241D6, C12.344(C2×D4), (C2×C6).81C24, C22⋊2(C2×Dic6), C6.45(C22×D4), C12.48D4⋊6C2, C6.12(C22×Q8), C22⋊C4.104D6, C2.13(Q8○D12), Dic3.16(C2×D4), (D4×Dic3).10C2, (C22×Dic6)⋊8C2, (C22×C4).217D6, (C2×C12).153C23, (C4×C12).144C22, Dic3.D4⋊6C2, (C6×D4).248C22, Dic3⋊C4.5C22, C2.14(C22×Dic6), C4⋊Dic3.197C22, C22.109(S3×C23), C23.172(C22×S3), (C22×C6).151C23, (C22×C12).76C22, (C4×Dic3).71C22, C6.D4.7C22, (C2×Dic3).199C23, (C2×Dic6).234C22, (C22×Dic3).90C22, (C2×C6)⋊1(C2×Q8), C2.18(C2×S3×D4), (C3×C4⋊C4).317C22, (C2×C4).152(C22×S3), (C3×C22⋊C4).103C22, SmallGroup(192,1096)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 280 conjugacy classes, 123 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×13], C22, C22 [×4], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×20], D4 [×4], Q8 [×16], C23 [×2], Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×3], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×11], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×15], Dic6 [×4], Dic6 [×12], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C22×C6 [×2], C4×D4, C4×D4 [×2], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×4], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×Dic6 [×6], C2×Dic6 [×8], C22×Dic3 [×4], C22×C12 [×2], C6×D4, D4×Q8, C4×Dic6, C12⋊2Q8, Dic3.D4 [×4], C12⋊Q8 [×2], C12.48D4 [×2], D4×Dic3 [×2], D4×C12, C22×Dic6 [×2], D4×Dic6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×7], C22×D4, C22×Q8, 2- (1+4), C2×Dic6 [×6], S3×D4 [×2], S3×C23, D4×Q8, C22×Dic6, C2×S3×D4, Q8○D12, D4×Dic6
Generators and relations
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►On 96 points
(1 51 34 40)(2 52 35 41)(3 53 36 42)(4 54 25 43)(5 55 26 44)(6 56 27 45)(7 57 28 46)(8 58 29 47)(9 59 30 48)(10 60 31 37)(11 49 32 38)(12 50 33 39)(13 80 93 69)(14 81 94 70)(15 82 95 71)(16 83 96 72)(17 84 85 61)(18 73 86 62)(19 74 87 63)(20 75 88 64)(21 76 89 65)(22 77 90 66)(23 78 91 67)(24 79 92 68)
(1 46)(2 47)(3 48)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 73)(25 60)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(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 72 7 66)(2 71 8 65)(3 70 9 64)(4 69 10 63)(5 68 11 62)(6 67 12 61)(13 60 19 54)(14 59 20 53)(15 58 21 52)(16 57 22 51)(17 56 23 50)(18 55 24 49)(25 80 31 74)(26 79 32 73)(27 78 33 84)(28 77 34 83)(29 76 35 82)(30 75 36 81)(37 87 43 93)(38 86 44 92)(39 85 45 91)(40 96 46 90)(41 95 47 89)(42 94 48 88)
G:=sub<Sym(96)| (1,51,34,40)(2,52,35,41)(3,53,36,42)(4,54,25,43)(5,55,26,44)(6,56,27,45)(7,57,28,46)(8,58,29,47)(9,59,30,48)(10,60,31,37)(11,49,32,38)(12,50,33,39)(13,80,93,69)(14,81,94,70)(15,82,95,71)(16,83,96,72)(17,84,85,61)(18,73,86,62)(19,74,87,63)(20,75,88,64)(21,76,89,65)(22,77,90,66)(23,78,91,67)(24,79,92,68), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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), (1,72,7,66)(2,71,8,65)(3,70,9,64)(4,69,10,63)(5,68,11,62)(6,67,12,61)(13,60,19,54)(14,59,20,53)(15,58,21,52)(16,57,22,51)(17,56,23,50)(18,55,24,49)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,87,43,93)(38,86,44,92)(39,85,45,91)(40,96,46,90)(41,95,47,89)(42,94,48,88)>;
G:=Group( (1,51,34,40)(2,52,35,41)(3,53,36,42)(4,54,25,43)(5,55,26,44)(6,56,27,45)(7,57,28,46)(8,58,29,47)(9,59,30,48)(10,60,31,37)(11,49,32,38)(12,50,33,39)(13,80,93,69)(14,81,94,70)(15,82,95,71)(16,83,96,72)(17,84,85,61)(18,73,86,62)(19,74,87,63)(20,75,88,64)(21,76,89,65)(22,77,90,66)(23,78,91,67)(24,79,92,68), (1,46)(2,47)(3,48)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,73)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,85)(68,86)(69,87)(70,88)(71,89)(72,90), (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), (1,72,7,66)(2,71,8,65)(3,70,9,64)(4,69,10,63)(5,68,11,62)(6,67,12,61)(13,60,19,54)(14,59,20,53)(15,58,21,52)(16,57,22,51)(17,56,23,50)(18,55,24,49)(25,80,31,74)(26,79,32,73)(27,78,33,84)(28,77,34,83)(29,76,35,82)(30,75,36,81)(37,87,43,93)(38,86,44,92)(39,85,45,91)(40,96,46,90)(41,95,47,89)(42,94,48,88) );
G=PermutationGroup([(1,51,34,40),(2,52,35,41),(3,53,36,42),(4,54,25,43),(5,55,26,44),(6,56,27,45),(7,57,28,46),(8,58,29,47),(9,59,30,48),(10,60,31,37),(11,49,32,38),(12,50,33,39),(13,80,93,69),(14,81,94,70),(15,82,95,71),(16,83,96,72),(17,84,85,61),(18,73,86,62),(19,74,87,63),(20,75,88,64),(21,76,89,65),(22,77,90,66),(23,78,91,67),(24,79,92,68)], [(1,46),(2,47),(3,48),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,73),(25,60),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,85),(68,86),(69,87),(70,88),(71,89),(72,90)], [(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)], [(1,72,7,66),(2,71,8,65),(3,70,9,64),(4,69,10,63),(5,68,11,62),(6,67,12,61),(13,60,19,54),(14,59,20,53),(15,58,21,52),(16,57,22,51),(17,56,23,50),(18,55,24,49),(25,80,31,74),(26,79,32,73),(27,78,33,84),(28,77,34,83),(29,76,35,82),(30,75,36,81),(37,87,43,93),(38,86,44,92),(39,85,45,91),(40,96,46,90),(41,95,47,89),(42,94,48,88)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 3 | 0 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 3 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 6 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 7 |
G:=sub<GL(6,GF(13))| [1,8,0,0,0,0,3,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,3,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,0,6,1,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | D6 | D6 | Dic6 | 2- (1+4) | S3×D4 | Q8○D12 |
| kernel | D4×Dic6 | C4×Dic6 | C12⋊2Q8 | Dic3.D4 | C12⋊Q8 | C12.48D4 | D4×Dic3 | D4×C12 | C22×Dic6 | C4×D4 | Dic6 | C3×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 1 | 4 | 4 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_4\times Dic_6
% in TeX
G:=Group("D4xDic6"); // GroupNames label
G:=SmallGroup(192,1096);
// by ID
G=gap.SmallGroup(192,1096);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations