direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.8D4, 2- 1+4⋊4C6, C4≀C2⋊2C6, C8⋊C22⋊2C6, D4.8(C3×D4), C4.28(C6×D4), (C3×D4).42D4, (C2×C12).24D4, C4.4D4⋊1C6, (C3×Q8).42D4, Q8.13(C3×D4), C4.10D4⋊1C6, C12.389(C2×D4), C42.12(C2×C6), C22.15(C6×D4), C6.101C22≀C2, M4(2).1(C2×C6), (C4×C12).254C22, (C2×C12).610C23, (C6×D4).181C22, (C3×2- 1+4)⋊6C2, (C6×Q8).157C22, (C3×M4(2)).15C22, (C3×C4≀C2)⋊6C2, (C2×C4).5(C3×D4), (C3×C8⋊C22)⋊9C2, C4○D4.7(C2×C6), (C2×D4).6(C2×C6), (C2×Q8).4(C2×C6), (C2×C6).410(C2×D4), (C2×C4).5(C22×C6), C2.15(C3×C22≀C2), (C3×C4.10D4)⋊5C2, (C3×C4.4D4)⋊21C2, (C3×C4○D4).32C22, SmallGroup(192,887)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.8D4
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >
Subgroups: 274 in 146 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C4×C12, C3×C22⋊C4, C3×M4(2), C3×D8, C3×SD16, C6×D4, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, D4.8D4, C3×C4.10D4, C3×C4≀C2, C3×C4.4D4, C3×C8⋊C22, C3×2- 1+4, C3×D4.8D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, D4.8D4, C3×C22≀C2, C3×D4.8D4
►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 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 43 45 47)(42 48 46 44)
(1 48)(2 43)(3 42)(4 45)(5 44)(6 47)(7 46)(8 41)(9 20)(10 19)(11 22)(12 21)(13 24)(14 23)(15 18)(16 17)(25 39)(26 38)(27 33)(28 40)(29 35)(30 34)(31 37)(32 36)
(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 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)(41 45)(42 44)(46 48)
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), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (1,48)(2,43)(3,42)(4,45)(5,44)(6,47)(7,46)(8,41)(9,20)(10,19)(11,22)(12,21)(13,24)(14,23)(15,18)(16,17)(25,39)(26,38)(27,33)(28,40)(29,35)(30,34)(31,37)(32,36), (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)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48)>;
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), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,43,45,47)(42,48,46,44), (1,48)(2,43)(3,42)(4,45)(5,44)(6,47)(7,46)(8,41)(9,20)(10,19)(11,22)(12,21)(13,24)(14,23)(15,18)(16,17)(25,39)(26,38)(27,33)(28,40)(29,35)(30,34)(31,37)(32,36), (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)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)(41,45)(42,44)(46,48) );
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)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,35,37,39),(34,40,38,36),(41,43,45,47),(42,48,46,44)], [(1,48),(2,43),(3,42),(4,45),(5,44),(6,47),(7,46),(8,41),(9,20),(10,19),(11,22),(12,21),(13,24),(14,23),(15,18),(16,17),(25,39),(26,38),(27,33),(28,40),(29,35),(30,34),(31,37),(32,36)], [(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),(9,11),(12,16),(13,15),(17,19),(20,24),(21,23),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38),(41,45),(42,44),(46,48)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | ··· | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C3×D4 | C3×D4 | C3×D4 | D4.8D4 | C3×D4.8D4 |
| kernel | C3×D4.8D4 | C3×C4.10D4 | C3×C4≀C2 | C3×C4.4D4 | C3×C8⋊C22 | C3×2- 1+4 | D4.8D4 | C4.10D4 | C4≀C2 | C4.4D4 | C8⋊C22 | 2- 1+4 | C2×C12 | C3×D4 | C3×Q8 | C2×C4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of C3×D4.8D4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 0 | 27 | 0 |
| 0 | 46 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,1,0,0,72,0,0,0,0,0,0,1,0,0,72,0],[0,0,0,46,0,0,46,0,0,27,0,0,27,0,0,0],[0,0,0,72,0,0,72,0,1,0,0,0,0,72,0,0],[1,0,0,0,0,72,0,0,0,0,0,72,0,0,72,0] >;
C3×D4.8D4 in GAP, Magma, Sage, TeX
C_3\times D_4._8D_4
% in TeX
G:=Group("C3xD4.8D4"); // GroupNames label
G:=SmallGroup(192,887);
// by ID
G=gap.SmallGroup(192,887);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,520,4204,2111,1068,172,3036]);
// 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=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=b^2*d^3>;
// generators/relations