direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8.7D6, C24.65D6, (S3×C8)⋊5C6, D4⋊S3⋊4C6, (S3×C24)⋊8C2, C24⋊C2⋊6C6, C8.11(S3×C6), C3⋊Q16⋊2C6, D4.5(S3×C6), D6.2(C3×D4), C6.33(C6×D4), D4⋊2S3⋊3C6, C24.11(C2×C6), Q8⋊3S3⋊5C6, (C3×SD16)⋊7S3, (C3×SD16)⋊4C6, SD16⋊3(C3×S3), D12.3(C2×C6), (C3×D4).28D6, (S3×C6).26D4, C6.193(S3×D4), Q8.12(S3×C6), (C3×Q8).49D6, C32⋊20(C4○D8), C12.7(C22×C6), Dic6.3(C2×C6), (C3×C24).38C22, (C3×C12).78C23, (C3×Dic3).50D4, Dic3.13(C3×D4), (C32×SD16)⋊6C2, (S3×C12).50C22, C12.158(C22×S3), (C3×D12).27C22, (C3×Dic6).26C22, (D4×C32).15C22, (Q8×C32).12C22, C4.7(S3×C2×C6), C3⋊3(C3×C4○D8), C3⋊C8.6(C2×C6), C2.21(C3×S3×D4), (C3×D4⋊S3)⋊10C2, (C3×D4).5(C2×C6), (C3×C3⋊Q16)⋊9C2, (C3×C24⋊C2)⋊14C2, (C3×Q8).7(C2×C6), (C3×D4⋊2S3)⋊6C2, (C4×S3).10(C2×C6), (C3×Q8⋊3S3)⋊5C2, (C3×C6).221(C2×D4), (C3×C3⋊C8).34C22, SmallGroup(288,687)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8.7D6
G = < a,b,c,d,e | a3=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >
Subgroups: 338 in 134 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, SD16, Q16, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, S3×C8, C24⋊C2, D4⋊S3, C3⋊Q16, C2×C24, C3×D8, C3×SD16, C3×SD16, C3×Q16, D4⋊2S3, Q8⋊3S3, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C6×Dic3, C3×C3⋊D4, D4×C32, Q8×C32, Q8.7D6, C3×C4○D8, S3×C24, C3×C24⋊C2, C3×D4⋊S3, C3×C3⋊Q16, C32×SD16, C3×D4⋊2S3, C3×Q8⋊3S3, C3×Q8.7D6
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C4○D8, S3×C6, S3×D4, C6×D4, S3×C2×C6, Q8.7D6, C3×C4○D8, C3×S3×D4, C3×Q8.7D6
►On 48 points
(1 3 2)(4 6 5)(7 8 9)(10 11 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 45 47)(44 46 48)
(1 17 5 14)(2 15 6 18)(3 13 4 16)(7 22 11 19)(8 20 12 23)(9 24 10 21)(25 28 46 43)(26 44 47 29)(27 30 48 45)(31 34 38 41)(32 42 39 35)(33 36 40 37)
(1 29 5 44)(2 27 6 48)(3 25 4 46)(7 31 11 38)(8 35 12 42)(9 33 10 40)(13 43 16 28)(14 26 17 47)(15 45 18 30)(19 34 22 41)(20 39 23 32)(21 36 24 37)
(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 24)(2 23 6 20)(3 19 4 22)(7 13 11 16)(8 15 12 18)(9 17 10 14)(25 31 46 38)(26 37 47 36)(27 35 48 42)(28 41 43 34)(29 33 44 40)(30 39 45 32)
G:=sub<Sym(48)| (1,3,2)(4,6,5)(7,8,9)(10,11,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,17,5,14)(2,15,6,18)(3,13,4,16)(7,22,11,19)(8,20,12,23)(9,24,10,21)(25,28,46,43)(26,44,47,29)(27,30,48,45)(31,34,38,41)(32,42,39,35)(33,36,40,37), (1,29,5,44)(2,27,6,48)(3,25,4,46)(7,31,11,38)(8,35,12,42)(9,33,10,40)(13,43,16,28)(14,26,17,47)(15,45,18,30)(19,34,22,41)(20,39,23,32)(21,36,24,37), (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), (1,21,5,24)(2,23,6,20)(3,19,4,22)(7,13,11,16)(8,15,12,18)(9,17,10,14)(25,31,46,38)(26,37,47,36)(27,35,48,42)(28,41,43,34)(29,33,44,40)(30,39,45,32)>;
G:=Group( (1,3,2)(4,6,5)(7,8,9)(10,11,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,45,47)(44,46,48), (1,17,5,14)(2,15,6,18)(3,13,4,16)(7,22,11,19)(8,20,12,23)(9,24,10,21)(25,28,46,43)(26,44,47,29)(27,30,48,45)(31,34,38,41)(32,42,39,35)(33,36,40,37), (1,29,5,44)(2,27,6,48)(3,25,4,46)(7,31,11,38)(8,35,12,42)(9,33,10,40)(13,43,16,28)(14,26,17,47)(15,45,18,30)(19,34,22,41)(20,39,23,32)(21,36,24,37), (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), (1,21,5,24)(2,23,6,20)(3,19,4,22)(7,13,11,16)(8,15,12,18)(9,17,10,14)(25,31,46,38)(26,37,47,36)(27,35,48,42)(28,41,43,34)(29,33,44,40)(30,39,45,32) );
G=PermutationGroup([[(1,3,2),(4,6,5),(7,8,9),(10,11,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,45,47),(44,46,48)], [(1,17,5,14),(2,15,6,18),(3,13,4,16),(7,22,11,19),(8,20,12,23),(9,24,10,21),(25,28,46,43),(26,44,47,29),(27,30,48,45),(31,34,38,41),(32,42,39,35),(33,36,40,37)], [(1,29,5,44),(2,27,6,48),(3,25,4,46),(7,31,11,38),(8,35,12,42),(9,33,10,40),(13,43,16,28),(14,26,17,47),(15,45,18,30),(19,34,22,41),(20,39,23,32),(21,36,24,37)], [(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)], [(1,21,5,24),(2,23,6,20),(3,19,4,22),(7,13,11,16),(8,15,12,18),(9,17,10,14),(25,31,46,38),(26,37,47,36),(27,35,48,42),(28,41,43,34),(29,33,44,40),(30,39,45,32)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12K | 12L | 12M | 12N | 12O | 12P | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 12 | 12 | 2 | 2 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 8 | 8 | 8 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
63 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 | 4 | 4 | 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 | C4○D8 | S3×C6 | S3×C6 | S3×C6 | C3×C4○D8 | S3×D4 | Q8.7D6 | C3×S3×D4 | C3×Q8.7D6 |
| kernel | C3×Q8.7D6 | S3×C24 | C3×C24⋊C2 | C3×D4⋊S3 | C3×C3⋊Q16 | C32×SD16 | C3×D4⋊2S3 | C3×Q8⋊3S3 | Q8.7D6 | S3×C8 | C24⋊C2 | D4⋊S3 | C3⋊Q16 | C3×SD16 | D4⋊2S3 | Q8⋊3S3 | C3×SD16 | C3×Dic3 | S3×C6 | C24 | C3×D4 | C3×Q8 | SD16 | Dic3 | D6 | C32 | C8 | D4 | Q8 | C3 | C6 | C3 | C2 | 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 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×Q8.7D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 71 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 61 | 12 | 0 | 0 |
| 67 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 27 | 19 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 0 | 64 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,71,72,0,0,0,0,1,0,0,0,0,1],[61,67,0,0,12,12,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,72,0,0,0,0,64,0,0,0,0,8],[27,0,0,0,19,46,0,0,0,0,0,64,0,0,8,0] >;
C3×Q8.7D6 in GAP, Magma, Sage, TeX
C_3\times Q_8._7D_6
% in TeX
G:=Group("C3xQ8.7D6"); // GroupNames label
G:=SmallGroup(288,687);
// by ID
G=gap.SmallGroup(288,687);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,1094,303,268,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^-1*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations