metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4:4Dic6, C42.45D6, C12.38SD16, (C3xD4):4Q8, (C4xD4).3S3, C4:C4.239D6, C12:C8:18C2, C3:4(D4:2Q8), (D4xC12).3C2, (C2xC12).58D4, C12.26(C2xQ8), C12:2Q8:15C2, (C2xD4).186D6, C4.10(C2xDic6), C6.51(C2xSD16), C12.46(C4oD4), C4.60(C4oD12), C2.7(D4:D6), C4.13(D4.S3), (C4xC12).79C22, C12.Q8:30C2, D4:Dic3.8C2, C6.62(C22:Q8), C6.107(C8:C22), (C2xC12).333C23, (C6xD4).228C22, C4:Dic3.137C22, C2.13(C12.48D4), C2.6(C2xD4.S3), (C2xC6).464(C2xD4), (C2xC3:C8).90C22, (C2xC4).244(C3:D4), (C3xC4:C4).270C22, (C2xC4).433(C22xS3), C22.147(C2xC3:D4), SmallGroup(192,567)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.38SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a-1, ac=ca, cbc=a6b3 >
Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C22xC4, C2xD4, C2xQ8, C3:C8, Dic6, C2xDic3, C2xC12, C2xC12, C3xD4, C3xD4, C22xC6, D4:C4, C4:C8, C4.Q8, C4xD4, C4:Q8, C2xC3:C8, C4:Dic3, C4:Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, C22xC12, C6xD4, D4:2Q8, C12:C8, C12.Q8, D4:Dic3, C12:2Q8, D4xC12, C12.38SD16
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, C2xSD16, C8:C22, D4.S3, C2xDic6, C4oD12, C2xC3:D4, D4:2Q8, C12.48D4, C2xD4.S3, D4:D6, C12.38SD16
►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 83 86 36 46 68 14 58)(2 82 87 35 47 67 15 57)(3 81 88 34 48 66 16 56)(4 80 89 33 37 65 17 55)(5 79 90 32 38 64 18 54)(6 78 91 31 39 63 19 53)(7 77 92 30 40 62 20 52)(8 76 93 29 41 61 21 51)(9 75 94 28 42 72 22 50)(10 74 95 27 43 71 23 49)(11 73 96 26 44 70 24 60)(12 84 85 25 45 69 13 59)
(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 73)(33 74)(34 75)(35 76)(36 77)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 61)(58 62)(59 63)(60 64)
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), (1,83,86,36,46,68,14,58)(2,82,87,35,47,67,15,57)(3,81,88,34,48,66,16,56)(4,80,89,33,37,65,17,55)(5,79,90,32,38,64,18,54)(6,78,91,31,39,63,19,53)(7,77,92,30,40,62,20,52)(8,76,93,29,41,61,21,51)(9,75,94,28,42,72,22,50)(10,74,95,27,43,71,23,49)(11,73,96,26,44,70,24,60)(12,84,85,25,45,69,13,59), (13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,73)(33,74)(34,75)(35,76)(36,77)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,61)(58,62)(59,63)(60,64)>;
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), (1,83,86,36,46,68,14,58)(2,82,87,35,47,67,15,57)(3,81,88,34,48,66,16,56)(4,80,89,33,37,65,17,55)(5,79,90,32,38,64,18,54)(6,78,91,31,39,63,19,53)(7,77,92,30,40,62,20,52)(8,76,93,29,41,61,21,51)(9,75,94,28,42,72,22,50)(10,74,95,27,43,71,23,49)(11,73,96,26,44,70,24,60)(12,84,85,25,45,69,13,59), (13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,73)(33,74)(34,75)(35,76)(36,77)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,61)(58,62)(59,63)(60,64) );
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)], [(1,83,86,36,46,68,14,58),(2,82,87,35,47,67,15,57),(3,81,88,34,48,66,16,56),(4,80,89,33,37,65,17,55),(5,79,90,32,38,64,18,54),(6,78,91,31,39,63,19,53),(7,77,92,30,40,62,20,52),(8,76,93,29,41,61,21,51),(9,75,94,28,42,72,22,50),(10,74,95,27,43,71,23,49),(11,73,96,26,44,70,24,60),(12,84,85,25,45,69,13,59)], [(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,73),(33,74),(34,75),(35,76),(36,77),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,61),(58,62),(59,63),(60,64)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | SD16 | C4oD4 | C3:D4 | Dic6 | C4oD12 | C8:C22 | D4.S3 | D4:D6 |
| kernel | C12.38SD16 | C12:C8 | C12.Q8 | D4:Dic3 | C12:2Q8 | D4xC12 | C4xD4 | C2xC12 | C3xD4 | C42 | C4:C4 | C2xD4 | C12 | C12 | C2xC4 | D4 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of C12.38SD16 ►in GL4(F73) generated by
| 3 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 42 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 66 | 72 |
G:=sub<GL(4,GF(73))| [3,0,0,0,0,49,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,1,0,0,0,0,0,0,42,0,0,40,12],[1,0,0,0,0,1,0,0,0,0,1,66,0,0,0,72] >;
C12.38SD16 in GAP, Magma, Sage, TeX
C_{12}._{38}{\rm SD}_{16} % in TeX
G:=Group("C12.38SD16"); // GroupNames label
G:=SmallGroup(192,567);
// by ID
G=gap.SmallGroup(192,567);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,253,120,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^6*b^3>;
// generators/relations