direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3xD4oD8, C24.48C23, C12.85C24, 2+ 1+4:8C6, C8oD4:7C6, C4oD8:4C6, D8:7(C2xC6), (C2xD8):12C6, (C6xD8):26C2, C8:C22:4C6, Q16:7(C2xC6), C4.45(C6xD4), SD16:4(C2xC6), D4.11(C3xD4), (C3xD4).45D4, C4.8(C23xC6), Q8.16(C3xD4), (C3xQ8).45D4, C22.7(C6xD4), (C2xC24):23C22, C12.406(C2xD4), (C6xD4):40C22, (C3xD8):21C22, M4(2):6(C2xC6), C8.10(C22xC6), D4.5(C22xC6), Q8.9(C22xC6), (C3xQ16):21C22, (C3xD4).38C23, C6.206(C22xD4), (C3xQ8).39C23, (C2xC12).687C23, (C3xSD16):20C22, (C3x2+ 1+4):9C2, (C3xM4(2)):27C22, (C2xC8):4(C2xC6), C2.30(D4xC2xC6), C4oD4:2(C2xC6), (C3xC8oD4):8C2, (C2xD4):7(C2xC6), (C3xC4oD8):11C2, (C3xC8:C22):11C2, (C2xC6).184(C2xD4), (C3xC4oD4):14C22, (C2xC4).48(C22xC6), SmallGroup(192,1465)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD4oD8
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >
Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C12, C12, C12, C2xC6, C2xC6, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xD4, C4oD4, C4oD4, C4oD4, C24, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C3xQ8, C22xC6, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, C2xC24, C3xM4(2), C3xD8, C3xSD16, C3xQ16, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, C3xC4oD4, D4oD8, C3xC8oD4, C6xD8, C3xC4oD8, C3xC8:C22, C3x2+ 1+4, C3xD4oD8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C24, C3xD4, C22xC6, C22xD4, C6xD4, C23xC6, D4oD8, D4xC2xC6, C3xD4oD8
►On 48 points
(1 35 19)(2 36 20)(3 37 21)(4 38 22)(5 39 23)(6 40 24)(7 33 17)(8 34 18)(9 41 26)(10 42 27)(11 43 28)(12 44 29)(13 45 30)(14 46 31)(15 47 32)(16 48 25)
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)
(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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(33 37)(34 36)(38 40)(42 48)(43 47)(44 46)
G:=sub<Sym(48)| (1,35,19)(2,36,20)(3,37,21)(4,38,22)(5,39,23)(6,40,24)(7,33,17)(8,34,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,34,29,38)(26,35,30,39)(27,36,31,40)(28,37,32,33), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(33,37)(34,36)(38,40)(42,48)(43,47)(44,46)>;
G:=Group( (1,35,19)(2,36,20)(3,37,21)(4,38,22)(5,39,23)(6,40,24)(7,33,17)(8,34,18)(9,41,26)(10,42,27)(11,43,28)(12,44,29)(13,45,30)(14,46,31)(15,47,32)(16,48,25), (1,45,5,41)(2,46,6,42)(3,47,7,43)(4,48,8,44)(9,19,13,23)(10,20,14,24)(11,21,15,17)(12,22,16,18)(25,34,29,38)(26,35,30,39)(27,36,31,40)(28,37,32,33), (9,13)(10,14)(11,15)(12,16)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(44,48), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(33,37)(34,36)(38,40)(42,48)(43,47)(44,46) );
G=PermutationGroup([[(1,35,19),(2,36,20),(3,37,21),(4,38,22),(5,39,23),(6,40,24),(7,33,17),(8,34,18),(9,41,26),(10,42,27),(11,43,28),(12,44,29),(13,45,30),(14,46,31),(15,47,32),(16,48,25)], [(1,45,5,41),(2,46,6,42),(3,47,7,43),(4,48,8,44),(9,19,13,23),(10,20,14,24),(11,21,15,17),(12,22,16,18),(25,34,29,38),(26,35,30,39),(27,36,31,40),(28,37,32,33)], [(9,13),(10,14),(11,15),(12,16),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48)], [(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)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(17,21),(18,20),(22,24),(25,27),(28,32),(29,31),(33,37),(34,36),(38,40),(42,48),(43,47),(44,46)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6T | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3xD4 | C3xD4 | D4oD8 | C3xD4oD8 |
| kernel | C3xD4oD8 | C3xC8oD4 | C6xD8 | C3xC4oD8 | C3xC8:C22 | C3x2+ 1+4 | D4oD8 | C8oD4 | C2xD8 | C4oD8 | C8:C22 | 2+ 1+4 | C3xD4 | C3xQ8 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 2 | 2 | 6 | 6 | 12 | 4 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C3xD4oD8 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 6 | 3 | 5 |
| 3 | 6 | 5 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 3 | 2 |
| 6 | 0 | 1 | 4 |
| 0 | 5 | 4 | 5 |
| 0 | 4 | 5 | 5 |
| 0 | 6 | 6 | 5 |
| 5 | 0 | 5 | 1 |
| 1 | 5 | 2 | 1 |
| 1 | 6 | 2 | 5 |
| 5 | 5 | 1 | 1 |
| 0 | 4 | 5 | 0 |
| 4 | 6 | 1 | 0 |
| 4 | 5 | 2 | 0 |
| 6 | 1 | 4 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,3,1,1,6,6,0,1,3,5,0,3,5,1,1,2],[6,0,0,0,0,5,4,6,1,4,5,6,4,5,5,5],[5,1,1,5,0,5,6,5,5,2,2,1,1,1,5,1],[0,4,4,6,4,6,5,1,5,1,2,4,0,0,0,6] >;
C3xD4oD8 in GAP, Magma, Sage, TeX
C_3\times D_4\circ D_8
% in TeX
G:=Group("C3xD4oD8"); // GroupNames label
G:=SmallGroup(192,1465);
// by ID
G=gap.SmallGroup(192,1465);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,745,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=e^2=1,d^4=b^2,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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations