direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D12, C42.127D6, C6.662- (1+4), C3⋊2(D4×Q8), C4⋊3(S3×Q8), (C3×Q8)⋊9D4, D6⋊5(C2×Q8), C12⋊8(C2×Q8), (C4×Q8)⋊12S3, C4⋊C4.294D6, (Q8×C12)⋊10C2, C12.56(C2×D4), C4.24(C2×D12), C4.D12⋊17C2, C12⋊2Q8⋊27C2, (C4×D12).20C2, (C2×Q8).227D6, C6.18(C22×D4), C6.29(C22×Q8), (C2×C6).119C24, C2.23(Q8○D12), C2.20(C22×D12), (C4×C12).171C22, (C2×C12).169C23, D6⋊C4.100C22, (C6×Q8).219C22, (C2×D12).288C22, C4⋊Dic3.305C22, C22.140(S3×C23), (C2×Dic3).53C23, (C22×S3).178C23, (C2×Dic6).148C22, (C2×S3×Q8)⋊3C2, C2.12(C2×S3×Q8), (S3×C2×C4).71C22, (C3×C4⋊C4).347C22, (C2×C4).583(C22×S3), SmallGroup(192,1134)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 680 in 280 conjugacy classes, 123 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C4 [×9], C22, C22 [×8], S3 [×4], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×18], D4 [×4], Q8 [×4], Q8 [×12], C23 [×2], Dic3 [×6], C12 [×8], C12 [×3], D6 [×4], D6 [×4], C2×C6, C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×6], C2×D4, C2×Q8, C2×Q8 [×14], Dic6 [×12], C4×S3 [×12], D12 [×4], C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8 [×6], C4⋊Q8 [×3], C22×Q8 [×2], C4⋊Dic3 [×9], D6⋊C4 [×6], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×6], S3×C2×C4 [×6], C2×D12, S3×Q8 [×8], C6×Q8, D4×Q8, C12⋊2Q8 [×3], C4×D12 [×3], C4.D12 [×6], Q8×C12, C2×S3×Q8 [×2], Q8×D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C22×Q8, 2- (1+4), C2×D12 [×6], S3×Q8 [×2], S3×C23, D4×Q8, C22×D12, C2×S3×Q8, Q8○D12, Q8×D12
Generators and relations
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 96 points
(1 74 43 23)(2 75 44 24)(3 76 45 13)(4 77 46 14)(5 78 47 15)(6 79 48 16)(7 80 37 17)(8 81 38 18)(9 82 39 19)(10 83 40 20)(11 84 41 21)(12 73 42 22)(25 85 67 52)(26 86 68 53)(27 87 69 54)(28 88 70 55)(29 89 71 56)(30 90 72 57)(31 91 61 58)(32 92 62 59)(33 93 63 60)(34 94 64 49)(35 95 65 50)(36 96 66 51)
(1 54 43 87)(2 55 44 88)(3 56 45 89)(4 57 46 90)(5 58 47 91)(6 59 48 92)(7 60 37 93)(8 49 38 94)(9 50 39 95)(10 51 40 96)(11 52 41 85)(12 53 42 86)(13 29 76 71)(14 30 77 72)(15 31 78 61)(16 32 79 62)(17 33 80 63)(18 34 81 64)(19 35 82 65)(20 36 83 66)(21 25 84 67)(22 26 73 68)(23 27 74 69)(24 28 75 70)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 31)(26 30)(27 29)(32 36)(33 35)(37 39)(40 48)(41 47)(42 46)(43 45)(50 60)(51 59)(52 58)(53 57)(54 56)(61 67)(62 66)(63 65)(68 72)(69 71)(73 77)(74 76)(78 84)(79 83)(80 82)(85 91)(86 90)(87 89)(92 96)(93 95)
G:=sub<Sym(96)| (1,74,43,23)(2,75,44,24)(3,76,45,13)(4,77,46,14)(5,78,47,15)(6,79,48,16)(7,80,37,17)(8,81,38,18)(9,82,39,19)(10,83,40,20)(11,84,41,21)(12,73,42,22)(25,85,67,52)(26,86,68,53)(27,87,69,54)(28,88,70,55)(29,89,71,56)(30,90,72,57)(31,91,61,58)(32,92,62,59)(33,93,63,60)(34,94,64,49)(35,95,65,50)(36,96,66,51), (1,54,43,87)(2,55,44,88)(3,56,45,89)(4,57,46,90)(5,58,47,91)(6,59,48,92)(7,60,37,93)(8,49,38,94)(9,50,39,95)(10,51,40,96)(11,52,41,85)(12,53,42,86)(13,29,76,71)(14,30,77,72)(15,31,78,61)(16,32,79,62)(17,33,80,63)(18,34,81,64)(19,35,82,65)(20,36,83,66)(21,25,84,67)(22,26,73,68)(23,27,74,69)(24,28,75,70), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,39)(40,48)(41,47)(42,46)(43,45)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,72)(69,71)(73,77)(74,76)(78,84)(79,83)(80,82)(85,91)(86,90)(87,89)(92,96)(93,95)>;
G:=Group( (1,74,43,23)(2,75,44,24)(3,76,45,13)(4,77,46,14)(5,78,47,15)(6,79,48,16)(7,80,37,17)(8,81,38,18)(9,82,39,19)(10,83,40,20)(11,84,41,21)(12,73,42,22)(25,85,67,52)(26,86,68,53)(27,87,69,54)(28,88,70,55)(29,89,71,56)(30,90,72,57)(31,91,61,58)(32,92,62,59)(33,93,63,60)(34,94,64,49)(35,95,65,50)(36,96,66,51), (1,54,43,87)(2,55,44,88)(3,56,45,89)(4,57,46,90)(5,58,47,91)(6,59,48,92)(7,60,37,93)(8,49,38,94)(9,50,39,95)(10,51,40,96)(11,52,41,85)(12,53,42,86)(13,29,76,71)(14,30,77,72)(15,31,78,61)(16,32,79,62)(17,33,80,63)(18,34,81,64)(19,35,82,65)(20,36,83,66)(21,25,84,67)(22,26,73,68)(23,27,74,69)(24,28,75,70), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,31)(26,30)(27,29)(32,36)(33,35)(37,39)(40,48)(41,47)(42,46)(43,45)(50,60)(51,59)(52,58)(53,57)(54,56)(61,67)(62,66)(63,65)(68,72)(69,71)(73,77)(74,76)(78,84)(79,83)(80,82)(85,91)(86,90)(87,89)(92,96)(93,95) );
G=PermutationGroup([(1,74,43,23),(2,75,44,24),(3,76,45,13),(4,77,46,14),(5,78,47,15),(6,79,48,16),(7,80,37,17),(8,81,38,18),(9,82,39,19),(10,83,40,20),(11,84,41,21),(12,73,42,22),(25,85,67,52),(26,86,68,53),(27,87,69,54),(28,88,70,55),(29,89,71,56),(30,90,72,57),(31,91,61,58),(32,92,62,59),(33,93,63,60),(34,94,64,49),(35,95,65,50),(36,96,66,51)], [(1,54,43,87),(2,55,44,88),(3,56,45,89),(4,57,46,90),(5,58,47,91),(6,59,48,92),(7,60,37,93),(8,49,38,94),(9,50,39,95),(10,51,40,96),(11,52,41,85),(12,53,42,86),(13,29,76,71),(14,30,77,72),(15,31,78,61),(16,32,79,62),(17,33,80,63),(18,34,81,64),(19,35,82,65),(20,36,83,66),(21,25,84,67),(22,26,73,68),(23,27,74,69),(24,28,75,70)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,31),(26,30),(27,29),(32,36),(33,35),(37,39),(40,48),(41,47),(42,46),(43,45),(50,60),(51,59),(52,58),(53,57),(54,56),(61,67),(62,66),(63,65),(68,72),(69,71),(73,77),(74,76),(78,84),(79,83),(80,82),(85,91),(86,90),(87,89),(92,96),(93,95)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 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 | 9 | 10 |
| 0 | 0 | 0 | 0 | 10 | 4 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 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(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,10,0,0,0,0,10,4],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | D6 | D12 | 2- (1+4) | S3×Q8 | Q8○D12 |
| kernel | Q8×D12 | C12⋊2Q8 | C4×D12 | C4.D12 | Q8×C12 | C2×S3×Q8 | C4×Q8 | D12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 4 | 4 | 3 | 3 | 1 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\times D_{12} % in TeX
G:=Group("Q8xD12"); // GroupNames label
G:=SmallGroup(192,1134);
// by ID
G=gap.SmallGroup(192,1134);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations