direct product, metabelian, supersoluble, monomial
Aliases: C2xC12:S3, C6:1D12, C12:6D6, C62.33C22, (C3xC6):5D4, (C2xC12):3S3, (C6xC12):4C2, C3:2(C2xD12), C32:9(C2xD4), (C2xC6).38D6, (C3xC12):6C22, C6.32(C22xS3), (C3xC6).31C23, C4:2(C2xC3:S3), (C2xC4):2(C3:S3), (C22xC3:S3):3C2, (C2xC3:S3):5C22, C2.4(C22xC3:S3), C22.10(C2xC3:S3), SmallGroup(144,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC12:S3
G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 642 in 162 conjugacy classes, 59 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C2xC4, D4, C23, C32, C12, D6, C2xC6, C2xD4, C3:S3, C3xC6, C3xC6, D12, C2xC12, C22xS3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C2xD12, C12:S3, C6xC12, C22xC3:S3, C2xC12:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, D12, C22xS3, C2xC3:S3, C2xD12, C12:S3, C22xC3:S3, C2xC12:S3
►On 72 points
(1 68)(2 69)(3 70)(4 71)(5 72)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 35)(14 36)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 33)(24 34)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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)
(1 49 16)(2 50 17)(3 51 18)(4 52 19)(5 53 20)(6 54 21)(7 55 22)(8 56 23)(9 57 24)(10 58 13)(11 59 14)(12 60 15)(25 67 42)(26 68 43)(27 69 44)(28 70 45)(29 71 46)(30 72 47)(31 61 48)(32 62 37)(33 63 38)(34 64 39)(35 65 40)(36 66 41)
(1 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 72)(10 71)(11 70)(12 69)(13 46)(14 45)(15 44)(16 43)(17 42)(18 41)(19 40)(20 39)(21 38)(22 37)(23 48)(24 47)(25 50)(26 49)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)
G:=sub<Sym(72)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,49,16)(2,50,17)(3,51,18)(4,52,19)(5,53,20)(6,54,21)(7,55,22)(8,56,23)(9,57,24)(10,58,13)(11,59,14)(12,60,15)(25,67,42)(26,68,43)(27,69,44)(28,70,45)(29,71,46)(30,72,47)(31,61,48)(32,62,37)(33,63,38)(34,64,39)(35,65,40)(36,66,41), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,50)(26,49)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)>;
G:=Group( (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,35)(14,36)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(21,31)(22,32)(23,33)(24,34)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,49,16)(2,50,17)(3,51,18)(4,52,19)(5,53,20)(6,54,21)(7,55,22)(8,56,23)(9,57,24)(10,58,13)(11,59,14)(12,60,15)(25,67,42)(26,68,43)(27,69,44)(28,70,45)(29,71,46)(30,72,47)(31,61,48)(32,62,37)(33,63,38)(34,64,39)(35,65,40)(36,66,41), (1,68)(2,67)(3,66)(4,65)(5,64)(6,63)(7,62)(8,61)(9,72)(10,71)(11,70)(12,69)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,48)(24,47)(25,50)(26,49)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51) );
G=PermutationGroup([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,35),(14,36),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(21,31),(22,32),(23,33),(24,34),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(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)], [(1,49,16),(2,50,17),(3,51,18),(4,52,19),(5,53,20),(6,54,21),(7,55,22),(8,56,23),(9,57,24),(10,58,13),(11,59,14),(12,60,15),(25,67,42),(26,68,43),(27,69,44),(28,70,45),(29,71,46),(30,72,47),(31,61,48),(32,62,37),(33,63,38),(34,64,39),(35,65,40),(36,66,41)], [(1,68),(2,67),(3,66),(4,65),(5,64),(6,63),(7,62),(8,61),(9,72),(10,71),(11,70),(12,69),(13,46),(14,45),(15,44),(16,43),(17,42),(18,41),(19,40),(20,39),(21,38),(22,37),(23,48),(24,47),(25,50),(26,49),(27,60),(28,59),(29,58),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51)]])
C2xC12:S3 is a maximal subgroup of
C12.70D12 C6.17D24 C62.113D4 C62.84D4 C12.19D12 (C6xC12):C4 D12:18D6 C12.28D12 Dic3:5D12 C62.67C23 C12:7D12 Dic3:3D12 C12:D12 D6:5D12 C12:4D12 C122:6C2 C62:12D4 C62.228C23 C62.237C23 C62.238C23 C12:3D12 C24:3D6 C62:19D4 C62.258C23 C62.262C23 C62.73D4 C2xS3xD12 D12:27D6 C2xD4xC3:S3 C62.154C23
C2xC12:S3 is a maximal quotient of
C12:6Dic6 C12:4D12 C122:6C2 C62:12D4 C62.69D4 C12:3D12 C12.31D12 C24.78D6 C24:3D6 C24.5D6 C62:19D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D12 |
| kernel | C2xC12:S3 | C12:S3 | C6xC12 | C22xC3:S3 | C2xC12 | C3xC6 | C12 | C2xC6 | C6 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 8 | 4 | 16 |
Matrix representation of C2xC12:S3 ►in GL5(F13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 6 | 3 | 0 | 0 |
| 0 | 10 | 3 | 0 | 0 |
| 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 10 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,6,10,0,0,0,3,3,0,0,0,0,0,6,10,0,0,0,3,3],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
C2xC12:S3 in GAP, Magma, Sage, TeX
C_2\times C_{12}\rtimes S_3 % in TeX
G:=Group("C2xC12:S3"); // GroupNames label
G:=SmallGroup(144,170);
// by ID
G=gap.SmallGroup(144,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,218,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations