direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8.13D6, C62.67D4, D4⋊S3⋊7C6, C4○D12⋊4C6, D4.S3⋊7C6, D4.8(S3×C6), C3⋊Q16⋊7C6, C6.60(C6×D4), Q8⋊2S3⋊7C6, (C3×D4).48D6, C12.66(C3×D4), (C3×Q8).73D6, Q8.18(S3×C6), D12.12(C2×C6), (C2×C12).333D6, (C3×C12).174D4, C32⋊24(C4○D8), (C3×C12).89C23, C12.18(C22×C6), Dic6.11(C2×C6), C12.149(C3⋊D4), C12.169(C22×S3), (C6×C12).132C22, (C3×D12).41C22, (C3×Dic6).41C22, (D4×C32).24C22, (Q8×C32).25C22, (C2×C3⋊C8)⋊8C6, (C6×C3⋊C8)⋊14C2, C3⋊5(C3×C4○D8), C4.18(S3×C2×C6), C4○D4⋊6(C3×S3), (C3×C4○D4)⋊6C6, C3⋊C8.10(C2×C6), (C3×D4⋊S3)⋊15C2, (C3×C4○D4)⋊11S3, (C3×C4○D12)⋊8C2, (C2×C4).59(S3×C6), (C3×D4).8(C2×C6), (C2×C6).10(C3×D4), C4.32(C3×C3⋊D4), C2.24(C6×C3⋊D4), (C2×C12).43(C2×C6), (C3×D4.S3)⋊15C2, (C3×C3⋊Q16)⋊15C2, (C3×C6).268(C2×D4), (C32×C4○D4)⋊2C2, C6.161(C2×C3⋊D4), (C3×C3⋊C8).41C22, (C3×Q8).20(C2×C6), C22.1(C3×C3⋊D4), (C3×Q8⋊2S3)⋊15C2, (C2×C6).29(C3⋊D4), SmallGroup(288,721)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.13D6
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, cd=dc, ece-1=b-1c, ede-1=d5 >
Subgroups: 322 in 144 conjugacy classes, 58 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×C12, C3×C12, S3×C6, C62, C62, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×Q16, C4○D12, C3×C4○D4, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, D4×C32, D4×C32, Q8×C32, Q8.13D6, C3×C4○D8, C6×C3⋊C8, C3×D4⋊S3, C3×D4.S3, C3×Q8⋊2S3, C3×C3⋊Q16, C3×C4○D12, C32×C4○D4, C3×Q8.13D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C4○D8, S3×C6, C2×C3⋊D4, C6×D4, C3×C3⋊D4, S3×C2×C6, Q8.13D6, C3×C4○D8, C6×C3⋊D4, C3×Q8.13D6
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)
(1 14 7 20)(2 15 8 21)(3 16 9 22)(4 17 10 23)(5 18 11 24)(6 19 12 13)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)
(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 33 7 27)(2 26 8 32)(3 31 9 25)(4 36 10 30)(5 29 11 35)(6 34 12 28)(13 40 19 46)(14 45 20 39)(15 38 21 44)(16 43 22 37)(17 48 23 42)(18 41 24 47)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,14,7,20)(2,15,8,21)(3,16,9,22)(4,17,10,23)(5,18,11,24)(6,19,12,13)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45), (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), (1,33,7,27)(2,26,8,32)(3,31,9,25)(4,36,10,30)(5,29,11,35)(6,34,12,28)(13,40,19,46)(14,45,20,39)(15,38,21,44)(16,43,22,37)(17,48,23,42)(18,41,24,47)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48), (1,14,7,20)(2,15,8,21)(3,16,9,22)(4,17,10,23)(5,18,11,24)(6,19,12,13)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,37,34,43)(29,38,35,44)(30,39,36,45), (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), (1,33,7,27)(2,26,8,32)(3,31,9,25)(4,36,10,30)(5,29,11,35)(6,34,12,28)(13,40,19,46)(14,45,20,39)(15,38,21,44)(16,43,22,37)(17,48,23,42)(18,41,24,47) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48)], [(1,14,7,20),(2,15,8,21),(3,16,9,22),(4,17,10,23),(5,18,11,24),(6,19,12,13),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,37,34,43),(29,38,35,44),(30,39,36,45)], [(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)], [(1,33,7,27),(2,26,8,32),(3,31,9,25),(4,36,10,30),(5,29,11,35),(6,34,12,28),(13,40,19,46),(14,45,20,39),(15,38,21,44),(16,43,22,37),(17,48,23,42),(18,41,24,47)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6G | 6H | ··· | 6R | 6S | 6T | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12W | 12X | 12Y | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 12 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | D6 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | C4○D8 | S3×C6 | S3×C6 | S3×C6 | C3×C3⋊D4 | C3×C3⋊D4 | C3×C4○D8 | Q8.13D6 | C3×Q8.13D6 |
| kernel | C3×Q8.13D6 | C6×C3⋊C8 | C3×D4⋊S3 | C3×D4.S3 | C3×Q8⋊2S3 | C3×C3⋊Q16 | C3×C4○D12 | C32×C4○D4 | Q8.13D6 | C2×C3⋊C8 | D4⋊S3 | D4.S3 | Q8⋊2S3 | C3⋊Q16 | C4○D12 | C3×C4○D4 | C3×C4○D4 | C3×C12 | C62 | C2×C12 | C3×D4 | C3×Q8 | C4○D4 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | D4 | Q8 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3×Q8.13D6 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 0 | 64 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 63 |
| 0 | 0 | 22 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[8,0,0,0,0,64,0,0,0,0,27,0,0,0,0,27],[0,8,0,0,64,0,0,0,0,0,0,22,0,0,63,0] >;
C3×Q8.13D6 in GAP, Magma, Sage, TeX
C_3\times Q_8._{13}D_6 % in TeX
G:=Group("C3xQ8.13D6"); // GroupNames label
G:=SmallGroup(288,721);
// by ID
G=gap.SmallGroup(288,721);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,590,2524,648,102,9414]);
// 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,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^5>;
// generators/relations