direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4.10D4, 2- 1+4.2C6, C4≀C2⋊4C6, C4⋊Q8⋊1C6, C4.30(C6×D4), (C3×D4).44D4, D4.10(C3×D4), (C2×C12).25D4, C8.C22⋊2C6, (C3×Q8).44D4, Q8.15(C3×D4), C4.10D4⋊2C6, C12.391(C2×D4), C42.13(C2×C6), C22.17(C6×D4), C6.103C22≀C2, M4(2).2(C2×C6), (C4×C12).255C22, (C2×C12).612C23, (C6×Q8).158C22, (C3×2- 1+4).2C2, (C3×M4(2)).16C22, (C3×C4≀C2)⋊8C2, (C3×C4⋊Q8)⋊22C2, (C2×C4).6(C3×D4), C4○D4.9(C2×C6), (C2×Q8).5(C2×C6), (C2×C6).412(C2×D4), (C3×C8.C22)⋊9C2, (C2×C4).7(C22×C6), C2.17(C3×C22≀C2), (C3×C4.10D4)⋊6C2, (C3×C4○D4).34C22, SmallGroup(192,889)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4.10D4
G = < a,b,c,d,e | a3=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >
Subgroups: 242 in 142 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, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4.10D4, C4≀C2, C4⋊Q8, C8.C22, 2- 1+4, C4×C12, C3×C4⋊C4, C3×M4(2), C3×SD16, C3×Q16, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, D4.10D4, C3×C4.10D4, C3×C4≀C2, C3×C4⋊Q8, C3×C8.C22, C3×2- 1+4, C3×D4.10D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, D4.10D4, C3×C22≀C2, C3×D4.10D4
►On 48 points
(1 34 45)(2 35 46)(3 36 47)(4 37 48)(5 38 41)(6 39 42)(7 40 43)(8 33 44)(9 19 32)(10 20 25)(11 21 26)(12 22 27)(13 23 28)(14 24 29)(15 17 30)(16 18 31)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)
(1 6)(2 5)(3 8)(4 7)(9 14)(10 13)(11 16)(12 15)(17 22)(18 21)(19 24)(20 23)(25 28)(26 31)(27 30)(29 32)(33 36)(34 39)(35 38)(37 40)(41 46)(42 45)(43 48)(44 47)
(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 21 5 17)(2 24 6 20)(3 19 7 23)(4 22 8 18)(9 43 13 47)(10 46 14 42)(11 41 15 45)(12 44 16 48)(25 35 29 39)(26 38 30 34)(27 33 31 37)(28 36 32 40)
G:=sub<Sym(48)| (1,34,45)(2,35,46)(3,36,47)(4,37,48)(5,38,41)(6,39,42)(7,40,43)(8,33,44)(9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48), (1,6)(2,5)(3,8)(4,7)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,28)(26,31)(27,30)(29,32)(33,36)(34,39)(35,38)(37,40)(41,46)(42,45)(43,48)(44,47), (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), (1,21,5,17)(2,24,6,20)(3,19,7,23)(4,22,8,18)(9,43,13,47)(10,46,14,42)(11,41,15,45)(12,44,16,48)(25,35,29,39)(26,38,30,34)(27,33,31,37)(28,36,32,40)>;
G:=Group( (1,34,45)(2,35,46)(3,36,47)(4,37,48)(5,38,41)(6,39,42)(7,40,43)(8,33,44)(9,19,32)(10,20,25)(11,21,26)(12,22,27)(13,23,28)(14,24,29)(15,17,30)(16,18,31), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48), (1,6)(2,5)(3,8)(4,7)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,28)(26,31)(27,30)(29,32)(33,36)(34,39)(35,38)(37,40)(41,46)(42,45)(43,48)(44,47), (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), (1,21,5,17)(2,24,6,20)(3,19,7,23)(4,22,8,18)(9,43,13,47)(10,46,14,42)(11,41,15,45)(12,44,16,48)(25,35,29,39)(26,38,30,34)(27,33,31,37)(28,36,32,40) );
G=PermutationGroup([[(1,34,45),(2,35,46),(3,36,47),(4,37,48),(5,38,41),(6,39,42),(7,40,43),(8,33,44),(9,19,32),(10,20,25),(11,21,26),(12,22,27),(13,23,28),(14,24,29),(15,17,30),(16,18,31)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28),(33,35,37,39),(34,40,38,36),(41,47,45,43),(42,44,46,48)], [(1,6),(2,5),(3,8),(4,7),(9,14),(10,13),(11,16),(12,15),(17,22),(18,21),(19,24),(20,23),(25,28),(26,31),(27,30),(29,32),(33,36),(34,39),(35,38),(37,40),(41,46),(42,45),(43,48),(44,47)], [(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)], [(1,21,5,17),(2,24,6,20),(3,19,7,23),(4,22,8,18),(9,43,13,47),(10,46,14,42),(11,41,15,45),(12,44,16,48),(25,35,29,39),(26,38,30,34),(27,33,31,37),(28,36,32,40)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | ··· | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | 12R | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 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 | D4.10D4 | C3×D4.10D4 |
| kernel | C3×D4.10D4 | C3×C4.10D4 | C3×C4≀C2 | C3×C4⋊Q8 | C3×C8.C22 | C3×2- 1+4 | D4.10D4 | C4.10D4 | C4≀C2 | C4⋊Q8 | 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×D4.10D4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 3 | 1 | 0 |
| 1 | 1 | 0 | 4 |
| 6 | 5 | 5 | 2 |
| 1 | 4 | 5 | 6 |
| 2 | 3 | 2 | 6 |
| 6 | 5 | 4 | 5 |
| 2 | 6 | 5 | 5 |
| 4 | 5 | 5 | 2 |
| 4 | 3 | 3 | 6 |
| 2 | 4 | 1 | 4 |
| 0 | 5 | 5 | 6 |
| 1 | 1 | 1 | 1 |
| 0 | 1 | 5 | 0 |
| 6 | 4 | 4 | 2 |
| 0 | 2 | 2 | 1 |
| 2 | 2 | 1 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,6,1,3,1,5,4,1,0,5,5,0,4,2,6],[2,6,2,4,3,5,6,5,2,4,5,5,6,5,5,2],[4,2,0,1,3,4,5,1,3,1,5,1,6,4,6,1],[0,6,0,2,1,4,2,2,5,4,2,1,0,2,1,1] >;
C3×D4.10D4 in GAP, Magma, Sage, TeX
C_3\times D_4._{10}D_4 % in TeX
G:=Group("C3xD4.10D4"); // GroupNames label
G:=SmallGroup(192,889);
// by ID
G=gap.SmallGroup(192,889);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,520,4204,2111,1068,172,3036]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=1,d^4=e^2=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^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations