direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C4○D8, C12○D8, D8⋊3C6, C12○Q16, Q16⋊3C6, C12○SD16, SD16⋊3C6, C12.69D4, C12.47C23, C24.28C22, C4○(C3×D8), (C2×C8)⋊4C6, C12○(C3×D8), C4○(C3×Q16), (C2×C24)⋊9C2, C4○D4⋊3C6, (C3×D8)⋊7C2, C8.6(C2×C6), C12○(C3×Q16), C4○(C3×SD16), (C3×Q16)⋊7C2, C12○(C3×SD16), D4.2(C2×C6), C2.14(C6×D4), C4.20(C3×D4), (C2×C6).11D4, C6.77(C2×D4), Q8.5(C2×C6), (C3×SD16)⋊7C2, C4.4(C22×C6), C22.1(C3×D4), (C3×D4).12C22, (C3×Q8).13C22, (C2×C12).132C22, (C3×C4○D4)⋊6C2, (C2×C4).28(C2×C6), SmallGroup(96,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4○D8
G = < a,b,c,d | a3=b4=d2=1, c4=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c3 >
Subgroups: 92 in 62 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C12, C12, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C4○D8, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×C4○D4, C3×C4○D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C4○D8, C6×D4, C3×C4○D8
►On 48 points
(1 12 19)(2 13 20)(3 14 21)(4 15 22)(5 16 23)(6 9 24)(7 10 17)(8 11 18)(25 36 48)(26 37 41)(27 38 42)(28 39 43)(29 40 44)(30 33 45)(31 34 46)(32 35 47)
(1 41 5 45)(2 42 6 46)(3 43 7 47)(4 44 8 48)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 35 21 39)(18 36 22 40)(19 37 23 33)(20 38 24 34)
(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)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)
G:=sub<Sym(48)| (1,12,19)(2,13,20)(3,14,21)(4,15,22)(5,16,23)(6,9,24)(7,10,17)(8,11,18)(25,36,48)(26,37,41)(27,38,42)(28,39,43)(29,40,44)(30,33,45)(31,34,46)(32,35,47), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34), (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)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46)>;
G:=Group( (1,12,19)(2,13,20)(3,14,21)(4,15,22)(5,16,23)(6,9,24)(7,10,17)(8,11,18)(25,36,48)(26,37,41)(27,38,42)(28,39,43)(29,40,44)(30,33,45)(31,34,46)(32,35,47), (1,41,5,45)(2,42,6,46)(3,43,7,47)(4,44,8,48)(9,31,13,27)(10,32,14,28)(11,25,15,29)(12,26,16,30)(17,35,21,39)(18,36,22,40)(19,37,23,33)(20,38,24,34), (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)(9,15)(10,14)(11,13)(17,21)(18,20)(22,24)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(42,48)(43,47)(44,46) );
G=PermutationGroup([[(1,12,19),(2,13,20),(3,14,21),(4,15,22),(5,16,23),(6,9,24),(7,10,17),(8,11,18),(25,36,48),(26,37,41),(27,38,42),(28,39,43),(29,40,44),(30,33,45),(31,34,46),(32,35,47)], [(1,41,5,45),(2,42,6,46),(3,43,7,47),(4,44,8,48),(9,31,13,27),(10,32,14,28),(11,25,15,29),(12,26,16,30),(17,35,21,39),(18,36,22,40),(19,37,23,33),(20,38,24,34)], [(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),(9,15),(10,14),(11,13),(17,21),(18,20),(22,24),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(42,48),(43,47),(44,46)]])
C3×C4○D8 is a maximal subgroup of
D8⋊2Dic3 C24.41D4 Q16⋊D6 Q16.D6 D8.9D6 D8⋊5Dic3 D8⋊4Dic3 SD16⋊D6 D8⋊15D6 D8⋊11D6 D8.10D6
C3×C4○D8 is a maximal quotient of
C12×D8 C12×SD16 C12×Q16
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D8 |
| kernel | C3×C4○D8 | C2×C24 | C3×D8 | C3×SD16 | C3×Q16 | C3×C4○D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C12 | C2×C6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C3×C4○D8 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 46 | 0 |
| 0 | 46 |
| 16 | 57 |
| 16 | 16 |
| 1 | 0 |
| 0 | 72 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[46,0,0,46],[16,16,57,16],[1,0,0,72] >;
C3×C4○D8 in GAP, Magma, Sage, TeX
C_3\times C_4\circ D_8
% in TeX
G:=Group("C3xC4oD8"); // GroupNames label
G:=SmallGroup(96,182);
// by ID
G=gap.SmallGroup(96,182);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,230,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^3>;
// generators/relations