metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊6D12, C42.128D6, C6.112- (1+4), (C3×Q8)⋊10D4, (C4×Q8)⋊13S3, (C4×D12)⋊38C2, C4⋊C4.295D6, (Q8×C12)⋊11C2, C3⋊2(Q8⋊5D4), C12.57(C2×D4), C4.25(C2×D12), D6⋊13(C4○D4), C12⋊D4⋊17C2, C4.D12⋊18C2, D6⋊C4.6C22, (C2×Q8).228D6, C6.19(C22×D4), C42⋊7S3⋊20C2, (C2×C6).120C24, C2.21(C22×D12), (C2×C12).498C23, (C4×C12).172C22, (C6×Q8).220C22, (C2×D12).216C22, (C22×S3).45C23, C4⋊Dic3.306C22, C22.141(S3×C23), (C2×Dic3).54C23, C2.12(Q8.15D6), (C2×Dic6).149C22, (C2×S3×Q8)⋊4C2, C2.29(S3×C4○D4), (C2×Q8⋊3S3)⋊3C2, C6.145(C2×C4○D4), (S3×C2×C4).72C22, (C3×C4⋊C4).348C22, (C2×C4).168(C22×S3), SmallGroup(192,1135)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 776 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×6], C4 [×8], C22, C22 [×13], S3 [×5], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×4], Q8 [×6], C23 [×4], Dic3 [×4], C12 [×6], C12 [×4], D6 [×2], D6 [×11], C2×C6, C42 [×3], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic6 [×6], C4×S3 [×12], D12 [×12], C2×Dic3, C2×Dic3 [×3], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×3], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4⋊Dic3 [×3], D6⋊C4, D6⋊C4 [×9], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4 [×6], C2×D12 [×6], S3×Q8 [×4], Q8⋊3S3 [×4], C6×Q8, Q8⋊5D4, C4×D12 [×3], C42⋊7S3 [×3], C12⋊D4 [×3], C4.D12 [×3], Q8×C12, C2×S3×Q8, C2×Q8⋊3S3, Q8⋊6D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2- (1+4), C2×D12 [×6], S3×C23, Q8⋊5D4, C22×D12, Q8.15D6, S3×C4○D4, Q8⋊6D12
Generators and relations
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 96 points
(1 56 26 87)(2 57 27 88)(3 58 28 89)(4 59 29 90)(5 60 30 91)(6 49 31 92)(7 50 32 93)(8 51 33 94)(9 52 34 95)(10 53 35 96)(11 54 36 85)(12 55 25 86)(13 68 46 83)(14 69 47 84)(15 70 48 73)(16 71 37 74)(17 72 38 75)(18 61 39 76)(19 62 40 77)(20 63 41 78)(21 64 42 79)(22 65 43 80)(23 66 44 81)(24 67 45 82)
(1 75 26 72)(2 61 27 76)(3 77 28 62)(4 63 29 78)(5 79 30 64)(6 65 31 80)(7 81 32 66)(8 67 33 82)(9 83 34 68)(10 69 35 84)(11 73 36 70)(12 71 25 74)(13 52 46 95)(14 96 47 53)(15 54 48 85)(16 86 37 55)(17 56 38 87)(18 88 39 57)(19 58 40 89)(20 90 41 59)(21 60 42 91)(22 92 43 49)(23 50 44 93)(24 94 45 51)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)(25 35)(26 34)(27 33)(28 32)(29 31)(49 59)(50 58)(51 57)(52 56)(53 55)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 84)(72 83)(86 96)(87 95)(88 94)(89 93)(90 92)
G:=sub<Sym(96)| (1,56,26,87)(2,57,27,88)(3,58,28,89)(4,59,29,90)(5,60,30,91)(6,49,31,92)(7,50,32,93)(8,51,33,94)(9,52,34,95)(10,53,35,96)(11,54,36,85)(12,55,25,86)(13,68,46,83)(14,69,47,84)(15,70,48,73)(16,71,37,74)(17,72,38,75)(18,61,39,76)(19,62,40,77)(20,63,41,78)(21,64,42,79)(22,65,43,80)(23,66,44,81)(24,67,45,82), (1,75,26,72)(2,61,27,76)(3,77,28,62)(4,63,29,78)(5,79,30,64)(6,65,31,80)(7,81,32,66)(8,67,33,82)(9,83,34,68)(10,69,35,84)(11,73,36,70)(12,71,25,74)(13,52,46,95)(14,96,47,53)(15,54,48,85)(16,86,37,55)(17,56,38,87)(18,88,39,57)(19,58,40,89)(20,90,41,59)(21,60,42,91)(22,92,43,49)(23,50,44,93)(24,94,45,51), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,35)(26,34)(27,33)(28,32)(29,31)(49,59)(50,58)(51,57)(52,56)(53,55)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83)(86,96)(87,95)(88,94)(89,93)(90,92)>;
G:=Group( (1,56,26,87)(2,57,27,88)(3,58,28,89)(4,59,29,90)(5,60,30,91)(6,49,31,92)(7,50,32,93)(8,51,33,94)(9,52,34,95)(10,53,35,96)(11,54,36,85)(12,55,25,86)(13,68,46,83)(14,69,47,84)(15,70,48,73)(16,71,37,74)(17,72,38,75)(18,61,39,76)(19,62,40,77)(20,63,41,78)(21,64,42,79)(22,65,43,80)(23,66,44,81)(24,67,45,82), (1,75,26,72)(2,61,27,76)(3,77,28,62)(4,63,29,78)(5,79,30,64)(6,65,31,80)(7,81,32,66)(8,67,33,82)(9,83,34,68)(10,69,35,84)(11,73,36,70)(12,71,25,74)(13,52,46,95)(14,96,47,53)(15,54,48,85)(16,86,37,55)(17,56,38,87)(18,88,39,57)(19,58,40,89)(20,90,41,59)(21,60,42,91)(22,92,43,49)(23,50,44,93)(24,94,45,51), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,35)(26,34)(27,33)(28,32)(29,31)(49,59)(50,58)(51,57)(52,56)(53,55)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83)(86,96)(87,95)(88,94)(89,93)(90,92) );
G=PermutationGroup([(1,56,26,87),(2,57,27,88),(3,58,28,89),(4,59,29,90),(5,60,30,91),(6,49,31,92),(7,50,32,93),(8,51,33,94),(9,52,34,95),(10,53,35,96),(11,54,36,85),(12,55,25,86),(13,68,46,83),(14,69,47,84),(15,70,48,73),(16,71,37,74),(17,72,38,75),(18,61,39,76),(19,62,40,77),(20,63,41,78),(21,64,42,79),(22,65,43,80),(23,66,44,81),(24,67,45,82)], [(1,75,26,72),(2,61,27,76),(3,77,28,62),(4,63,29,78),(5,79,30,64),(6,65,31,80),(7,81,32,66),(8,67,33,82),(9,83,34,68),(10,69,35,84),(11,73,36,70),(12,71,25,74),(13,52,46,95),(14,96,47,53),(15,54,48,85),(16,86,37,55),(17,56,38,87),(18,88,39,57),(19,58,40,89),(20,90,41,59),(21,60,42,91),(22,92,43,49),(23,50,44,93),(24,94,45,51)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,38),(14,37),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39),(25,35),(26,34),(27,33),(28,32),(29,31),(49,59),(50,58),(51,57),(52,56),(53,55),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,84),(72,83),(86,96),(87,95),(88,94),(89,93),(90,92)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 6 | 10 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[6,3,0,0,10,3,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,1,1,0,0,0,0,1,0,0,0,0,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | 2- (1+4) | Q8.15D6 | S3×C4○D4 |
| kernel | Q8⋊6D12 | C4×D12 | C42⋊7S3 | C12⋊D4 | C4.D12 | Q8×C12 | C2×S3×Q8 | C2×Q8⋊3S3 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | D6 | Q8 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
Q_8\rtimes_6D_{12} % in TeX
G:=Group("Q8:6D12"); // GroupNames label
G:=SmallGroup(192,1135);
// by ID
G=gap.SmallGroup(192,1135);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,192,6278]);
// 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,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations