direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4○SD16, C24.49C23, C12.86C24, 2+ 1+4⋊9C6, 2- 1+4⋊5C6, C4○D8⋊5C6, C8○D4⋊8C6, D8⋊5(C2×C6), C8⋊C22⋊5C6, Q16⋊5(C2×C6), C4.46(C6×D4), SD16⋊6(C2×C6), (C2×SD16)⋊6C6, D4.12(C3×D4), (C3×D4).46D4, C8.C22⋊4C6, C4.9(C23×C6), (C3×Q8).46D4, Q8.17(C3×D4), C22.8(C6×D4), (C2×C24)⋊24C22, (C6×SD16)⋊17C2, C12.407(C2×D4), (C3×D8)⋊22C22, M4(2)⋊7(C2×C6), C8.13(C22×C6), (C6×Q8)⋊32C22, D4.6(C22×C6), (C3×Q16)⋊19C22, (C3×D4).39C23, C6.207(C22×D4), Q8.10(C22×C6), (C3×Q8).40C23, (C2×C12).688C23, (C3×SD16)⋊21C22, (C6×D4).226C22, (C3×2- 1+4)⋊7C2, (C3×M4(2))⋊28C22, (C3×2+ 1+4)⋊10C2, (C2×C8)⋊5(C2×C6), C2.31(D4×C2×C6), (C3×C8○D4)⋊9C2, C4○D4⋊3(C2×C6), (C2×Q8)⋊8(C2×C6), (C3×C4○D8)⋊12C2, (C3×C8⋊C22)⋊12C2, (C2×D4).39(C2×C6), (C2×C6).185(C2×D4), (C3×C4○D4)⋊15C22, (C3×C8.C22)⋊11C2, (C2×C4).49(C22×C6), SmallGroup(192,1466)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4○SD16
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=d3 >
Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C12, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C22×C6, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C2×C24, C3×M4(2), C3×D8, C3×SD16, C3×SD16, C3×Q16, C6×D4, C6×D4, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×C4○D4, D4○SD16, C3×C8○D4, C6×SD16, C3×C4○D8, C3×C8⋊C22, C3×C8.C22, C3×2+ 1+4, C3×2- 1+4, C3×D4○SD16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C24, C3×D4, C22×C6, C22×D4, C6×D4, C23×C6, D4○SD16, D4×C2×C6, C3×D4○SD16
►On 48 points
(1 33 18)(2 34 19)(3 35 20)(4 36 21)(5 37 22)(6 38 23)(7 39 24)(8 40 17)(9 42 29)(10 43 30)(11 44 31)(12 45 32)(13 46 25)(14 47 26)(15 48 27)(16 41 28)
(1 47 5 43)(2 48 6 44)(3 41 7 45)(4 42 8 46)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)(25 36 29 40)(26 37 30 33)(27 38 31 34)(28 39 32 35)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(34 36)(35 39)(38 40)(41 45)(42 48)(44 46)
G:=sub<Sym(48)| (1,33,18)(2,34,19)(3,35,20)(4,36,21)(5,37,22)(6,38,23)(7,39,24)(8,40,17)(9,42,29)(10,43,30)(11,44,31)(12,45,32)(13,46,25)(14,47,26)(15,48,27)(16,41,28), (1,47,5,43)(2,48,6,44)(3,41,7,45)(4,42,8,46)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,36,29,40)(26,37,30,33)(27,38,31,34)(28,39,32,35), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40)(41,45)(42,48)(44,46)>;
G:=Group( (1,33,18)(2,34,19)(3,35,20)(4,36,21)(5,37,22)(6,38,23)(7,39,24)(8,40,17)(9,42,29)(10,43,30)(11,44,31)(12,45,32)(13,46,25)(14,47,26)(15,48,27)(16,41,28), (1,47,5,43)(2,48,6,44)(3,41,7,45)(4,42,8,46)(9,17,13,21)(10,18,14,22)(11,19,15,23)(12,20,16,24)(25,36,29,40)(26,37,30,33)(27,38,31,34)(28,39,32,35), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(34,36)(35,39)(38,40)(41,45)(42,48)(44,46) );
G=PermutationGroup([[(1,33,18),(2,34,19),(3,35,20),(4,36,21),(5,37,22),(6,38,23),(7,39,24),(8,40,17),(9,42,29),(10,43,30),(11,44,31),(12,45,32),(13,46,25),(14,47,26),(15,48,27),(16,41,28)], [(1,47,5,43),(2,48,6,44),(3,41,7,45),(4,42,8,46),(9,17,13,21),(10,18,14,22),(11,19,15,23),(12,20,16,24),(25,36,29,40),(26,37,30,33),(27,38,31,34),(28,39,32,35)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(34,36),(35,39),(38,40),(41,45),(42,48),(44,46)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 8E | 12A | ··· | 12H | 12I | ··· | 12P | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | D4○SD16 | C3×D4○SD16 |
| kernel | C3×D4○SD16 | C3×C8○D4 | C6×SD16 | C3×C4○D8 | C3×C8⋊C22 | C3×C8.C22 | C3×2+ 1+4 | C3×2- 1+4 | D4○SD16 | C8○D4 | C2×SD16 | C4○D8 | C8⋊C22 | C8.C22 | 2+ 1+4 | 2- 1+4 | C3×D4 | C3×Q8 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C3×D4○SD16 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 6 | 67 |
| 0 | 0 | 6 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[6,6,0,0,67,6,0,0,0,0,6,6,0,0,67,6],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
C3×D4○SD16 in GAP, Magma, Sage, TeX
C_3\times D_4\circ {\rm SD}_{16} % in TeX
G:=Group("C3xD4oSD16"); // GroupNames label
G:=SmallGroup(192,1466);
// by ID
G=gap.SmallGroup(192,1466);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,745,6053,3036,124]);
// 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=d^3>;
// generators/relations