direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD8:S3, D8:9D6, C24:5C23, C12.2C24, Dic6:1C23, D12.1C23, (C2xC8):8D6, C3:C8:1C23, (C6xD8):11C2, (C2xD8):11S3, (C2xD4):28D6, C8:3(C22xS3), C4.40(S3xD4), C6:2(C8:C22), D4:S3:9C22, (C4xS3).14D4, D6.49(C2xD4), C12.77(C2xD4), (S3xD4):5C22, D4:2(C22xS3), (C3xD4):2C23, C4.2(S3xC23), (C2xC24):16C22, (C6xD4):19C22, (C3xD8):14C22, (C4xS3).1C23, D4.S3:7C22, C8:S3:12C22, C24:C2:13C22, D4:2S3:5C22, Dic3.54(C2xD4), (C22xS3).97D4, C6.103(C22xD4), C22.136(S3xD4), (C2xC12).519C23, (C2xDic3).191D4, (C2xDic6):36C22, (C2xD12).176C22, (C2xS3xD4):22C2, C3:2(C2xC8:C22), C2.76(C2xS3xD4), (C2xC8:S3):8C2, (C2xD4:S3):26C2, (C2xC3:C8):14C22, (C2xC24:C2):24C2, (C2xD4.S3):25C2, (C2xC6).392(C2xD4), (C2xD4:2S3):23C2, (S3xC2xC4).155C22, (C2xC4).609(C22xS3), SmallGroup(192,1314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD8:S3
G = < a,b,c,d,e | a2=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
Subgroups: 920 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC8, C2xC8, M4(2), D8, D8, SD16, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C24, C3:C8, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xD4, C22xS3, C22xS3, C22xC6, C2xM4(2), C2xD8, C2xD8, C2xSD16, C8:C22, C22xD4, C2xC4oD4, C8:S3, C24:C2, C2xC3:C8, D4:S3, D4.S3, C2xC24, C3xD8, C2xDic6, S3xC2xC4, C2xD12, S3xD4, S3xD4, D4:2S3, D4:2S3, C22xDic3, C2xC3:D4, C6xD4, S3xC23, C2xC8:C22, C2xC8:S3, C2xC24:C2, D8:S3, C2xD4:S3, C2xD4.S3, C6xD8, C2xS3xD4, C2xD4:2S3, C2xD8:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C24, C22xS3, C8:C22, C22xD4, S3xD4, S3xC23, C2xC8:C22, D8:S3, C2xS3xD4, C2xD8:S3
►On 48 points
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(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)
(1 7)(2 6)(3 5)(9 13)(10 12)(14 16)(18 24)(19 23)(20 22)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)
(1 34 48)(2 35 41)(3 36 42)(4 37 43)(5 38 44)(6 39 45)(7 40 46)(8 33 47)(9 23 28)(10 24 29)(11 17 30)(12 18 31)(13 19 32)(14 20 25)(15 21 26)(16 22 27)
(1 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 35)(10 40)(11 37)(12 34)(13 39)(14 36)(15 33)(16 38)(25 42)(26 47)(27 44)(28 41)(29 46)(30 43)(31 48)(32 45)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,7)(2,6)(3,5)(9,13)(10,12)(14,16)(18,24)(19,23)(20,22)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48), (1,34,48)(2,35,41)(3,36,42)(4,37,43)(5,38,44)(6,39,45)(7,40,46)(8,33,47)(9,23,28)(10,24,29)(11,17,30)(12,18,31)(13,19,32)(14,20,25)(15,21,26)(16,22,27), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,42)(26,47)(27,44)(28,41)(29,46)(30,43)(31,48)(32,45)>;
G:=Group( (1,22)(2,23)(3,24)(4,17)(5,18)(6,19)(7,20)(8,21)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (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), (1,7)(2,6)(3,5)(9,13)(10,12)(14,16)(18,24)(19,23)(20,22)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48), (1,34,48)(2,35,41)(3,36,42)(4,37,43)(5,38,44)(6,39,45)(7,40,46)(8,33,47)(9,23,28)(10,24,29)(11,17,30)(12,18,31)(13,19,32)(14,20,25)(15,21,26)(16,22,27), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(25,42)(26,47)(27,44)(28,41)(29,46)(30,43)(31,48)(32,45) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,17),(5,18),(6,19),(7,20),(8,21),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(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)], [(1,7),(2,6),(3,5),(9,13),(10,12),(14,16),(18,24),(19,23),(20,22),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(41,45),(42,44),(46,48)], [(1,34,48),(2,35,41),(3,36,42),(4,37,43),(5,38,44),(6,39,45),(7,40,46),(8,33,47),(9,23,28),(10,24,29),(11,17,30),(12,18,31),(13,19,32),(14,20,25),(15,21,26),(16,22,27)], [(1,18),(2,23),(3,20),(4,17),(5,22),(6,19),(7,24),(8,21),(9,35),(10,40),(11,37),(12,34),(13,39),(14,36),(15,33),(16,38),(25,42),(26,47),(27,44),(28,41),(29,46),(30,43),(31,48),(32,45)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C8:C22 | S3xD4 | S3xD4 | D8:S3 |
| kernel | C2xD8:S3 | C2xC8:S3 | C2xC24:C2 | D8:S3 | C2xD4:S3 | C2xD4.S3 | C6xD8 | C2xS3xD4 | C2xD4:2S3 | C2xD8 | C4xS3 | C2xDic3 | C22xS3 | C2xC8 | D8 | C2xD4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 4 |
Matrix representation of C2xD8:S3 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 62 |
| 0 | 0 | 0 | 0 | 11 | 42 |
| 0 | 0 | 11 | 22 | 11 | 22 |
| 0 | 0 | 51 | 62 | 51 | 62 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 71 | 0 | 72 | 0 |
| 0 | 0 | 0 | 71 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,0,0,11,51,0,0,0,0,22,62,0,0,31,11,11,51,0,0,62,42,22,62],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,71,0,0,0,0,1,0,71,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C2xD8:S3 in GAP, Magma, Sage, TeX
C_2\times D_8\rtimes S_3
% in TeX
G:=Group("C2xD8:S3"); // GroupNames label
G:=SmallGroup(192,1314);
// by ID
G=gap.SmallGroup(192,1314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations