metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12:1SD16, D12.18D4, C42.32D6, C4:C8:8S3, C4:3(C24:C2), C4.129(S3xD4), (C2xC8).128D6, C12:2Q8:11C2, (C4xD12).10C2, (C2xC4).131D12, C12.338(C2xD4), (C2xC12).120D4, C3:2(D4.D4), C6.10(C2xSD16), C6.37(C4:D4), (C4xC12).67C22, C2.Dic12:12C2, C12.327(C4oD4), C2.10(C12:D4), (C2xC24).138C22, (C2xC12).751C23, C4.43(Q8:3S3), C2.17(C8.D6), C22.114(C2xD12), C6.14(C8.C22), (C2xD12).194C22, C4:Dic3.272C22, (C2xDic6).14C22, (C3xC4:C8):10C2, (C2xC24:C2).5C2, C2.13(C2xC24:C2), (C2xC6).134(C2xD4), (C2xC4).696(C22xS3), SmallGroup(192,400)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a7, cac=a5, cbc=b3 >
Subgroups: 392 in 120 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C24, Dic6, C4xS3, D12, D12, C2xDic3, C2xC12, C22xS3, Q8:C4, C4:C8, C4xD4, C4:Q8, C2xSD16, C24:C2, C4:Dic3, C4:Dic3, D6:C4, C4xC12, C2xC24, C2xDic6, S3xC2xC4, C2xD12, D4.D4, C2.Dic12, C3xC4:C8, C12:2Q8, C4xD12, C2xC24:C2, C12:SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C4oD4, D12, C22xS3, C4:D4, C2xSD16, C8.C22, C24:C2, C2xD12, S3xD4, Q8:3S3, D4.D4, C12:D4, C2xC24:C2, C8.D6, C12:SD16
►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 64 34 59 47 16 91 79)(2 71 35 54 48 23 92 74)(3 66 36 49 37 18 93 81)(4 61 25 56 38 13 94 76)(5 68 26 51 39 20 95 83)(6 63 27 58 40 15 96 78)(7 70 28 53 41 22 85 73)(8 65 29 60 42 17 86 80)(9 72 30 55 43 24 87 75)(10 67 31 50 44 19 88 82)(11 62 32 57 45 14 89 77)(12 69 33 52 46 21 90 84)
(2 6)(3 11)(5 9)(8 12)(13 76)(14 81)(15 74)(16 79)(17 84)(18 77)(19 82)(20 75)(21 80)(22 73)(23 78)(24 83)(25 94)(26 87)(27 92)(28 85)(29 90)(30 95)(31 88)(32 93)(33 86)(34 91)(35 96)(36 89)(37 45)(39 43)(40 48)(42 46)(49 62)(50 67)(51 72)(52 65)(53 70)(54 63)(55 68)(56 61)(57 66)(58 71)(59 64)(60 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), (1,64,34,59,47,16,91,79)(2,71,35,54,48,23,92,74)(3,66,36,49,37,18,93,81)(4,61,25,56,38,13,94,76)(5,68,26,51,39,20,95,83)(6,63,27,58,40,15,96,78)(7,70,28,53,41,22,85,73)(8,65,29,60,42,17,86,80)(9,72,30,55,43,24,87,75)(10,67,31,50,44,19,88,82)(11,62,32,57,45,14,89,77)(12,69,33,52,46,21,90,84), (2,6)(3,11)(5,9)(8,12)(13,76)(14,81)(15,74)(16,79)(17,84)(18,77)(19,82)(20,75)(21,80)(22,73)(23,78)(24,83)(25,94)(26,87)(27,92)(28,85)(29,90)(30,95)(31,88)(32,93)(33,86)(34,91)(35,96)(36,89)(37,45)(39,43)(40,48)(42,46)(49,62)(50,67)(51,72)(52,65)(53,70)(54,63)(55,68)(56,61)(57,66)(58,71)(59,64)(60,69)>;
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), (1,64,34,59,47,16,91,79)(2,71,35,54,48,23,92,74)(3,66,36,49,37,18,93,81)(4,61,25,56,38,13,94,76)(5,68,26,51,39,20,95,83)(6,63,27,58,40,15,96,78)(7,70,28,53,41,22,85,73)(8,65,29,60,42,17,86,80)(9,72,30,55,43,24,87,75)(10,67,31,50,44,19,88,82)(11,62,32,57,45,14,89,77)(12,69,33,52,46,21,90,84), (2,6)(3,11)(5,9)(8,12)(13,76)(14,81)(15,74)(16,79)(17,84)(18,77)(19,82)(20,75)(21,80)(22,73)(23,78)(24,83)(25,94)(26,87)(27,92)(28,85)(29,90)(30,95)(31,88)(32,93)(33,86)(34,91)(35,96)(36,89)(37,45)(39,43)(40,48)(42,46)(49,62)(50,67)(51,72)(52,65)(53,70)(54,63)(55,68)(56,61)(57,66)(58,71)(59,64)(60,69) );
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)], [(1,64,34,59,47,16,91,79),(2,71,35,54,48,23,92,74),(3,66,36,49,37,18,93,81),(4,61,25,56,38,13,94,76),(5,68,26,51,39,20,95,83),(6,63,27,58,40,15,96,78),(7,70,28,53,41,22,85,73),(8,65,29,60,42,17,86,80),(9,72,30,55,43,24,87,75),(10,67,31,50,44,19,88,82),(11,62,32,57,45,14,89,77),(12,69,33,52,46,21,90,84)], [(2,6),(3,11),(5,9),(8,12),(13,76),(14,81),(15,74),(16,79),(17,84),(18,77),(19,82),(20,75),(21,80),(22,73),(23,78),(24,83),(25,94),(26,87),(27,92),(28,85),(29,90),(30,95),(31,88),(32,93),(33,86),(34,91),(35,96),(36,89),(37,45),(39,43),(40,48),(42,46),(49,62),(50,67),(51,72),(52,65),(53,70),(54,63),(55,68),(56,61),(57,66),(58,71),(59,64),(60,69)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C4oD4 | D12 | C24:C2 | C8.C22 | S3xD4 | Q8:3S3 | C8.D6 |
| kernel | C12:SD16 | C2.Dic12 | C3xC4:C8 | C12:2Q8 | C4xD12 | C2xC24:C2 | C4:C8 | D12 | C2xC12 | C42 | C2xC8 | C12 | C12 | C2xC4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of C12:SD16 ►in GL6(F73)
| 72 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 7 | 59 | 0 | 0 | 0 | 0 |
| 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 18 | 0 | 0 |
| 0 | 0 | 65 | 47 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 14 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,14,65,0,0,0,0,18,47,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,14,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C12:SD16 in GAP, Magma, Sage, TeX
C_{12}\rtimes {\rm SD}_{16} % in TeX
G:=Group("C12:SD16"); // GroupNames label
G:=SmallGroup(192,400);
// by ID
G=gap.SmallGroup(192,400);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^5,c*b*c=b^3>;
// generators/relations