direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.4D4, C24.93D4, C8○D4⋊6C6, (C2×D8)⋊8C6, (C6×D8)⋊22C2, C8⋊C22⋊3C6, D4.4(C3×D4), C8.19(C3×D4), C4.38(C6×D4), Q8.9(C3×D4), C8.C4⋊6C6, (C3×D4).29D4, C4.D4⋊4C6, (C3×Q8).29D4, C12.399(C2×D4), M4(2).4(C2×C6), C6.155(C4⋊D4), (C2×C24).201C22, (C2×C12).614C23, (C6×D4).194C22, (C3×M4(2)).48C22, (C3×C8○D4)⋊7C2, (C2×C8).25(C2×C6), (C3×C8⋊C22)⋊10C2, C4○D4.18(C2×C6), (C2×D4).17(C2×C6), (C3×C4.D4)⋊8C2, C2.24(C3×C4⋊D4), (C2×C4).9(C22×C6), (C3×C8.C4)⋊15C2, C22.7(C3×C4○D4), (C2×C6).116(C4○D4), (C3×C4○D4).56C22, SmallGroup(192,905)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.4D4
G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >
Subgroups: 226 in 108 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C24, C24, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C3×M4(2), C3×D8, C3×SD16, C6×D4, C3×C4○D4, D4.4D4, C3×C4.D4, C3×C8.C4, C3×C8○D4, C6×D8, C3×C8⋊C22, C3×D4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C6×D4, C3×C4○D4, D4.4D4, C3×C4⋊D4, C3×D4.4D4
►On 48 points
(1 33 23)(2 34 24)(3 35 17)(4 36 18)(5 37 19)(6 38 20)(7 39 21)(8 40 22)(9 48 28)(10 41 29)(11 42 30)(12 43 31)(13 44 32)(14 45 25)(15 46 26)(16 47 27)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(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)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)(25 26)(27 32)(28 31)(29 30)(33 40)(34 39)(35 38)(36 37)(41 42)(43 48)(44 47)(45 46)
G:=sub<Sym(48)| (1,33,23)(2,34,24)(3,35,17)(4,36,18)(5,37,19)(6,38,20)(7,39,21)(8,40,22)(9,48,28)(10,41,29)(11,42,30)(12,43,31)(13,44,32)(14,45,25)(15,46,26)(16,47,27), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,42)(43,48)(44,47)(45,46)>;
G:=Group( (1,33,23)(2,34,24)(3,35,17)(4,36,18)(5,37,19)(6,38,20)(7,39,21)(8,40,22)(9,48,28)(10,41,29)(11,42,30)(12,43,31)(13,44,32)(14,45,25)(15,46,26)(16,47,27), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,37)(26,38)(27,39)(28,40)(29,33)(30,34)(31,35)(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), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,32)(28,31)(29,30)(33,40)(34,39)(35,38)(36,37)(41,42)(43,48)(44,47)(45,46) );
G=PermutationGroup([[(1,33,23),(2,34,24),(3,35,17),(4,36,18),(5,37,19),(6,38,20),(7,39,21),(8,40,22),(9,48,28),(10,41,29),(11,42,30),(12,43,31),(13,44,32),(14,45,25),(15,46,26),(16,47,27)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,37),(26,38),(27,39),(28,40),(29,33),(30,34),(31,35),(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)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23),(25,26),(27,32),(28,31),(29,30),(33,40),(34,39),(35,38),(36,37),(41,42),(43,48),(44,47),(45,46)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×D4 | C3×C4○D4 | D4.4D4 | C3×D4.4D4 |
| kernel | C3×D4.4D4 | C3×C4.D4 | C3×C8.C4 | C3×C8○D4 | C6×D8 | C3×C8⋊C22 | D4.4D4 | C4.D4 | C8.C4 | C8○D4 | C2×D8 | C8⋊C22 | C24 | C3×D4 | C3×Q8 | C2×C6 | C8 | D4 | Q8 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C3×D4.4D4 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 6 | 5 | 1 |
| 3 | 0 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 1 | 6 | 3 | 1 |
| 1 | 2 | 5 | 3 |
| 4 | 0 | 2 | 2 |
| 6 | 6 | 0 | 2 |
| 6 | 1 | 4 | 6 |
| 2 | 0 | 2 | 6 |
| 6 | 2 | 5 | 6 |
| 6 | 1 | 5 | 2 |
| 2 | 2 | 6 | 6 |
| 2 | 4 | 2 | 5 |
| 6 | 4 | 5 | 2 |
| 6 | 0 | 1 | 5 |
| 2 | 5 | 6 | 0 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[1,4,6,6,2,0,6,1,5,2,0,4,3,2,2,6],[2,6,6,2,0,2,1,2,2,5,5,6,6,6,2,6],[2,6,6,2,4,4,0,5,2,5,1,6,5,2,5,0] >;
C3×D4.4D4 in GAP, Magma, Sage, TeX
C_3\times D_4._4D_4
% in TeX
G:=Group("C3xD4.4D4"); // GroupNames label
G:=SmallGroup(192,905);
// by ID
G=gap.SmallGroup(192,905);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,848,1094,4204,172,6053,1531,124]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=b^2*d^3>;
// generators/relations