direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8.11D6, C62.125D4, (C6×Q8)⋊6C6, (C6×Q8)⋊13S3, C3⋊Q16⋊5C6, C6.54(C6×D4), Q8⋊2S3⋊5C6, C4○D12.5C6, (C3×C12).89D4, C12.19(C3×D4), C4.Dic3⋊7C6, (C3×Q8).71D6, Q8.16(S3×C6), D12.10(C2×C6), (C2×C12).243D6, Dic6.9(C2×C6), C12.15(C22×C6), (C3×C12).86C23, C12.104(C3⋊D4), C12.166(C22×S3), (C6×C12).125C22, (C3×D12).39C22, C32⋊22(C8.C22), (C3×Dic6).39C22, (Q8×C32).23C22, (Q8×C3×C6)⋊2C2, C3⋊C8.3(C2×C6), C4.15(S3×C2×C6), (C2×Q8)⋊6(C3×S3), (C2×C4).18(S3×C6), (C2×C6).51(C3×D4), C4.17(C3×C3⋊D4), C2.18(C6×C3⋊D4), C3⋊4(C3×C8.C22), (C2×C12).36(C2×C6), (C3×C3⋊Q16)⋊13C2, (C3×C6).262(C2×D4), C6.155(C2×C3⋊D4), (C3×C3⋊C8).22C22, (C3×C4.Dic3)⋊6C2, (C3×Q8).18(C2×C6), (C3×Q8⋊2S3)⋊11C2, (C3×C4○D12).11C2, (C2×C6).64(C3⋊D4), C22.11(C3×C3⋊D4), SmallGroup(288,713)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.11D6
G = < a,b,c,d,e | a3=b4=1, c2=d6=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d5 >
Subgroups: 298 in 139 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C4.Dic3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C4○D12, C6×Q8, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, Q8×C32, Q8×C32, Q8.11D6, C3×C8.C22, C3×C4.Dic3, C3×Q8⋊2S3, C3×C3⋊Q16, C3×C4○D12, Q8×C3×C6, C3×Q8.11D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C8.C22, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, Q8.11D6, C3×C8.C22, C6×C3⋊D4, C3×Q8.11D6
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 16 19 22)(14 17 20 23)(15 18 21 24)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)
(1 17 7 23)(2 24 8 18)(3 19 9 13)(4 14 10 20)(5 21 11 15)(6 16 12 22)(25 37 31 43)(26 44 32 38)(27 39 33 45)(28 46 34 40)(29 41 35 47)(30 48 36 42)
(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 36 7 30)(2 29 8 35)(3 34 9 28)(4 27 10 33)(5 32 11 26)(6 25 12 31)(13 37 19 43)(14 42 20 48)(15 47 21 41)(16 40 22 46)(17 45 23 39)(18 38 24 44)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42), (1,17,7,23)(2,24,8,18)(3,19,9,13)(4,14,10,20)(5,21,11,15)(6,16,12,22)(25,37,31,43)(26,44,32,38)(27,39,33,45)(28,46,34,40)(29,41,35,47)(30,48,36,42), (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), (1,36,7,30)(2,29,8,35)(3,34,9,28)(4,27,10,33)(5,32,11,26)(6,25,12,31)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,16,19,22)(14,17,20,23)(15,18,21,24)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,46,43,40)(38,47,44,41)(39,48,45,42), (1,17,7,23)(2,24,8,18)(3,19,9,13)(4,14,10,20)(5,21,11,15)(6,16,12,22)(25,37,31,43)(26,44,32,38)(27,39,33,45)(28,46,34,40)(29,41,35,47)(30,48,36,42), (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), (1,36,7,30)(2,29,8,35)(3,34,9,28)(4,27,10,33)(5,32,11,26)(6,25,12,31)(13,37,19,43)(14,42,20,48)(15,47,21,41)(16,40,22,46)(17,45,23,39)(18,38,24,44) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,16,19,22),(14,17,20,23),(15,18,21,24),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,46,43,40),(38,47,44,41),(39,48,45,42)], [(1,17,7,23),(2,24,8,18),(3,19,9,13),(4,14,10,20),(5,21,11,15),(6,16,12,22),(25,37,31,43),(26,44,32,38),(27,39,33,45),(28,46,34,40),(29,41,35,47),(30,48,36,42)], [(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)], [(1,36,7,30),(2,29,8,35),(3,34,9,28),(4,27,10,33),(5,32,11,26),(6,25,12,31),(13,37,19,43),(14,42,20,48),(15,47,21,41),(16,40,22,46),(17,45,23,39),(18,38,24,44)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6M | 6N | 6O | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 1 | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 12 | 12 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | S3×C6 | C3×C3⋊D4 | C3×C3⋊D4 | C8.C22 | Q8.11D6 | C3×C8.C22 | C3×Q8.11D6 |
| kernel | C3×Q8.11D6 | C3×C4.Dic3 | C3×Q8⋊2S3 | C3×C3⋊Q16 | C3×C4○D12 | Q8×C3×C6 | Q8.11D6 | C4.Dic3 | Q8⋊2S3 | C3⋊Q16 | C4○D12 | C6×Q8 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C2×Q8 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | Q8 | C4 | C22 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×Q8.11D6 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 6 | 5 | 3 |
| 2 | 0 | 2 | 2 |
| 5 | 5 | 6 | 4 |
| 1 | 6 | 3 | 0 |
| 0 | 3 | 4 | 2 |
| 0 | 6 | 3 | 6 |
| 2 | 2 | 6 | 5 |
| 6 | 1 | 1 | 2 |
| 4 | 0 | 5 | 1 |
| 4 | 4 | 1 | 2 |
| 5 | 4 | 2 | 0 |
| 4 | 6 | 6 | 4 |
| 5 | 3 | 1 | 0 |
| 2 | 3 | 0 | 1 |
| 3 | 4 | 2 | 4 |
| 5 | 5 | 5 | 4 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,2,5,1,6,0,5,6,5,2,6,3,3,2,4,0],[0,0,2,6,3,6,2,1,4,3,6,1,2,6,5,2],[4,4,5,4,0,4,4,6,5,1,2,6,1,2,0,4],[5,2,3,5,3,3,4,5,1,0,2,5,0,1,4,4] >;
C3×Q8.11D6 in GAP, Magma, Sage, TeX
C_3\times Q_8._{11}D_6 % in TeX
G:=Group("C3xQ8.11D6"); // GroupNames label
G:=SmallGroup(288,713);
// by ID
G=gap.SmallGroup(288,713);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,555,268,2524,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=1,c^2=d^6=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^5>;
// generators/relations