direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8.15D6, C32⋊62- 1+4, C62.272C23, (S3×Q8)⋊7C6, C4○D12⋊6C6, (C6×Q8)⋊16S3, (C6×Q8)⋊11C6, Q8⋊3S3⋊7C6, (C3×Q8).75D6, Q8.20(S3×C6), D12.13(C2×C6), (C2×C12).255D6, C6.10(C23×C6), C6.78(S3×C23), (C3×C6).47C24, D6.5(C22×C6), (S3×C6).32C23, C12.24(C22×C6), Dic6.13(C2×C6), C3⋊1(C3×2- 1+4), (S3×C12).36C22, C12.175(C22×S3), (C6×C12).166C22, (C3×C12).125C23, (C3×D12).44C22, Dic3.6(C22×C6), (C3×Dic6).45C22, (C3×Dic3).33C23, (Q8×C32).31C22, C4.24(S3×C2×C6), (Q8×C3×C6)⋊11C2, (C3×S3×Q8)⋊11C2, (C2×Q8)⋊9(C3×S3), C22.7(S3×C2×C6), (C4×S3).5(C2×C6), (C2×C4).22(S3×C6), C3⋊D4.2(C2×C6), C2.11(S3×C22×C6), (C3×C4○D12)⋊16C2, (C2×C12).48(C2×C6), (C3×Q8).22(C2×C6), (C3×Q8⋊3S3)⋊11C2, (C2×C6).73(C22×C6), (C3×C3⋊D4).5C22, (C2×C6).165(C22×S3), SmallGroup(288,997)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.15D6
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=ece-1=b2c, ede-1=d5 >
Subgroups: 586 in 311 conjugacy classes, 170 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×Q8, C2×Q8, C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, 2- 1+4, C3×Dic3, C3×C12, S3×C6, C62, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, C6×Q8, C3×C4○D4, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, Q8×C32, Q8.15D6, C3×2- 1+4, C3×C4○D12, C3×S3×Q8, C3×Q8⋊3S3, Q8×C3×C6, C3×Q8.15D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, C3×S3, C22×S3, C22×C6, 2- 1+4, S3×C6, S3×C23, C23×C6, S3×C2×C6, Q8.15D6, C3×2- 1+4, S3×C22×C6, C3×Q8.15D6
►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 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(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 43 31 37)(26 38 32 44)(27 45 33 39)(28 40 34 46)(29 47 35 41)(30 42 36 48)
(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 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)
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), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,17,7,23)(2,24,8,18)(3,19,9,13)(4,14,10,20)(5,21,11,15)(6,16,12,22)(25,43,31,37)(26,38,32,44)(27,45,33,39)(28,40,34,46)(29,47,35,41)(30,42,36,48), (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)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41)>;
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), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,17,7,23)(2,24,8,18)(3,19,9,13)(4,14,10,20)(5,21,11,15)(6,16,12,22)(25,43,31,37)(26,38,32,44)(27,45,33,39)(28,40,34,46)(29,47,35,41)(30,42,36,48), (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)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41) );
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)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(1,17,7,23),(2,24,8,18),(3,19,9,13),(4,14,10,20),(5,21,11,15),(6,16,12,22),(25,43,31,37),(26,38,32,44),(27,45,33,39),(28,40,34,46),(29,47,35,41),(30,42,36,48)], [(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),(13,46,19,40),(14,39,20,45),(15,44,21,38),(16,37,22,43),(17,42,23,48),(18,47,24,41)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | ··· | 6M | 6N | ··· | 6U | 12A | ··· | 12L | 12M | ··· | 12AD | 12AE | ··· | 12AL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C3×S3 | S3×C6 | S3×C6 | 2- 1+4 | Q8.15D6 | C3×2- 1+4 | C3×Q8.15D6 |
| kernel | C3×Q8.15D6 | C3×C4○D12 | C3×S3×Q8 | C3×Q8⋊3S3 | Q8×C3×C6 | Q8.15D6 | C4○D12 | S3×Q8 | Q8⋊3S3 | C6×Q8 | C6×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C32 | C3 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 2 | 12 | 8 | 8 | 2 | 1 | 3 | 4 | 2 | 6 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×Q8.15D6 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 3 | 4 | 1 |
| 0 | 1 | 6 | 0 |
| 0 | 2 | 6 | 0 |
| 4 | 5 | 5 | 3 |
| 5 | 1 | 1 | 1 |
| 0 | 2 | 5 | 2 |
| 1 | 3 | 1 | 1 |
| 1 | 4 | 3 | 6 |
| 0 | 3 | 2 | 3 |
| 4 | 1 | 4 | 5 |
| 3 | 4 | 3 | 0 |
| 2 | 5 | 6 | 3 |
| 4 | 5 | 0 | 1 |
| 0 | 6 | 6 | 1 |
| 4 | 1 | 6 | 4 |
| 4 | 6 | 5 | 5 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,0,0,4,3,1,2,5,4,6,6,5,1,0,0,3],[5,0,1,1,1,2,3,4,1,5,1,3,1,2,1,6],[0,4,3,2,3,1,4,5,2,4,3,6,3,5,0,3],[4,0,4,4,5,6,1,6,0,6,6,5,1,1,4,5] >;
C3×Q8.15D6 in GAP, Magma, Sage, TeX
C_3\times Q_8._{15}D_6 % in TeX
G:=Group("C3xQ8.15D6"); // GroupNames label
G:=SmallGroup(288,997);
// by ID
G=gap.SmallGroup(288,997);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,9414]);
// 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=e*c*e^-1=b^2*c,e*d*e^-1=d^5>;
// generators/relations