metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:9Q8, C42.175D6, C6.832+ 1+4, C4:Q8:13S3, C4.19(S3xQ8), C4:C4.220D6, C3:8(D4:3Q8), D6.12(C2xQ8), C12.56(C2xQ8), D6:3Q8:37C2, C4.D12:44C2, (C4xDic6):53C2, (C4xD12).27C2, (C2xQ8).111D6, C6.50(C22xQ8), (C2xC6).274C24, C4.Dic6:44C2, Dic3:5D4.14C2, C12.137(C4oD4), C2.87(D4:6D6), (C4xC12).215C22, (C2xC12).107C23, D6:C4.153C22, C4.40(Q8:3S3), (C6xQ8).141C22, (C2xD12).273C22, Dic3:C4.62C22, C4:Dic3.253C22, C22.295(S3xC23), (C22xS3).235C23, (C2xDic6).303C22, (C2xDic3).145C23, (C4xDic3).163C22, (S3xC4:C4):45C2, C2.33(C2xS3xQ8), (C3xC4:Q8):16C2, C6.122(C2xC4oD4), (S3xC2xC4).147C22, C2.30(C2xQ8:3S3), (C3xC4:C4).217C22, (C2xC4).220(C22xS3), SmallGroup(192,1289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:9Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a6b, dcd-1=c-1 >
Subgroups: 544 in 228 conjugacy classes, 107 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C42, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C2xQ8, Dic6, C4xS3, D12, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C2xC4:C4, C4xD4, C4xQ8, C22:Q8, C42.C2, C4:Q8, C4xDic3, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, C6xQ8, D4:3Q8, C4xDic6, C4xD12, C4.Dic6, S3xC4:C4, Dic3:5D4, C4.D12, D6:3Q8, C3xC4:Q8, D12:9Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C4oD4, C24, C22xS3, C22xQ8, C2xC4oD4, 2+ 1+4, S3xQ8, Q8:3S3, S3xC23, D4:3Q8, D4:6D6, C2xS3xQ8, C2xQ8:3S3, D12:9Q8
►On 96 points
(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 17)(14 16)(18 24)(19 23)(20 22)(25 33)(26 32)(27 31)(28 30)(34 36)(37 41)(38 40)(42 48)(43 47)(44 46)(49 59)(50 58)(51 57)(52 56)(53 55)(61 65)(62 64)(66 72)(67 71)(68 70)(74 84)(75 83)(76 82)(77 81)(78 80)(85 95)(86 94)(87 93)(88 92)(89 91)
(1 25 38 65)(2 32 39 72)(3 27 40 67)(4 34 41 62)(5 29 42 69)(6 36 43 64)(7 31 44 71)(8 26 45 66)(9 33 46 61)(10 28 47 68)(11 35 48 63)(12 30 37 70)(13 55 77 85)(14 50 78 92)(15 57 79 87)(16 52 80 94)(17 59 81 89)(18 54 82 96)(19 49 83 91)(20 56 84 86)(21 51 73 93)(22 58 74 88)(23 53 75 95)(24 60 76 90)
(1 86 38 56)(2 87 39 57)(3 88 40 58)(4 89 41 59)(5 90 42 60)(6 91 43 49)(7 92 44 50)(8 93 45 51)(9 94 46 52)(10 95 47 53)(11 96 48 54)(12 85 37 55)(13 30 77 70)(14 31 78 71)(15 32 79 72)(16 33 80 61)(17 34 81 62)(18 35 82 63)(19 36 83 64)(20 25 84 65)(21 26 73 66)(22 27 74 67)(23 28 75 68)(24 29 76 69)
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,41)(38,40)(42,48)(43,47)(44,46)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,72)(67,71)(68,70)(74,84)(75,83)(76,82)(77,81)(78,80)(85,95)(86,94)(87,93)(88,92)(89,91), (1,25,38,65)(2,32,39,72)(3,27,40,67)(4,34,41,62)(5,29,42,69)(6,36,43,64)(7,31,44,71)(8,26,45,66)(9,33,46,61)(10,28,47,68)(11,35,48,63)(12,30,37,70)(13,55,77,85)(14,50,78,92)(15,57,79,87)(16,52,80,94)(17,59,81,89)(18,54,82,96)(19,49,83,91)(20,56,84,86)(21,51,73,93)(22,58,74,88)(23,53,75,95)(24,60,76,90), (1,86,38,56)(2,87,39,57)(3,88,40,58)(4,89,41,59)(5,90,42,60)(6,91,43,49)(7,92,44,50)(8,93,45,51)(9,94,46,52)(10,95,47,53)(11,96,48,54)(12,85,37,55)(13,30,77,70)(14,31,78,71)(15,32,79,72)(16,33,80,61)(17,34,81,62)(18,35,82,63)(19,36,83,64)(20,25,84,65)(21,26,73,66)(22,27,74,67)(23,28,75,68)(24,29,76,69)>;
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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,17)(14,16)(18,24)(19,23)(20,22)(25,33)(26,32)(27,31)(28,30)(34,36)(37,41)(38,40)(42,48)(43,47)(44,46)(49,59)(50,58)(51,57)(52,56)(53,55)(61,65)(62,64)(66,72)(67,71)(68,70)(74,84)(75,83)(76,82)(77,81)(78,80)(85,95)(86,94)(87,93)(88,92)(89,91), (1,25,38,65)(2,32,39,72)(3,27,40,67)(4,34,41,62)(5,29,42,69)(6,36,43,64)(7,31,44,71)(8,26,45,66)(9,33,46,61)(10,28,47,68)(11,35,48,63)(12,30,37,70)(13,55,77,85)(14,50,78,92)(15,57,79,87)(16,52,80,94)(17,59,81,89)(18,54,82,96)(19,49,83,91)(20,56,84,86)(21,51,73,93)(22,58,74,88)(23,53,75,95)(24,60,76,90), (1,86,38,56)(2,87,39,57)(3,88,40,58)(4,89,41,59)(5,90,42,60)(6,91,43,49)(7,92,44,50)(8,93,45,51)(9,94,46,52)(10,95,47,53)(11,96,48,54)(12,85,37,55)(13,30,77,70)(14,31,78,71)(15,32,79,72)(16,33,80,61)(17,34,81,62)(18,35,82,63)(19,36,83,64)(20,25,84,65)(21,26,73,66)(22,27,74,67)(23,28,75,68)(24,29,76,69) );
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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,17),(14,16),(18,24),(19,23),(20,22),(25,33),(26,32),(27,31),(28,30),(34,36),(37,41),(38,40),(42,48),(43,47),(44,46),(49,59),(50,58),(51,57),(52,56),(53,55),(61,65),(62,64),(66,72),(67,71),(68,70),(74,84),(75,83),(76,82),(77,81),(78,80),(85,95),(86,94),(87,93),(88,92),(89,91)], [(1,25,38,65),(2,32,39,72),(3,27,40,67),(4,34,41,62),(5,29,42,69),(6,36,43,64),(7,31,44,71),(8,26,45,66),(9,33,46,61),(10,28,47,68),(11,35,48,63),(12,30,37,70),(13,55,77,85),(14,50,78,92),(15,57,79,87),(16,52,80,94),(17,59,81,89),(18,54,82,96),(19,49,83,91),(20,56,84,86),(21,51,73,93),(22,58,74,88),(23,53,75,95),(24,60,76,90)], [(1,86,38,56),(2,87,39,57),(3,88,40,58),(4,89,41,59),(5,90,42,60),(6,91,43,49),(7,92,44,50),(8,93,45,51),(9,94,46,52),(10,95,47,53),(11,96,48,54),(12,85,37,55),(13,30,77,70),(14,31,78,71),(15,32,79,72),(16,33,80,61),(17,34,81,62),(18,35,82,63),(19,36,83,64),(20,25,84,65),(21,26,73,66),(22,27,74,67),(23,28,75,68),(24,29,76,69)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 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 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4oD4 | 2+ 1+4 | S3xQ8 | Q8:3S3 | D4:6D6 |
| kernel | D12:9Q8 | C4xDic6 | C4xD12 | C4.Dic6 | S3xC4:C4 | Dic3:5D4 | C4.D12 | D6:3Q8 | C3xC4:Q8 | C4:Q8 | D12 | C42 | C4:C4 | C2xQ8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12:9Q8 ►in GL6(F13)
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 |
| 12 | 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 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 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 |
| 8 | 10 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,3,9] >;
D12:9Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_9Q_8 % in TeX
G:=Group("D12:9Q8"); // GroupNames label
G:=SmallGroup(192,1289);
// by ID
G=gap.SmallGroup(192,1289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,570,185,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations