direct product, metabelian, supersoluble, monomial
Aliases: C2×D6.6D6, Dic6⋊24D6, C62.133C23, (C4×S3)⋊13D6, C6⋊1(C4○D12), C6.8(S3×C23), (C3×C6).8C24, C6⋊1(Q8⋊3S3), (C6×Dic6)⋊21C2, (C2×Dic6)⋊15S3, (C2×C12).285D6, (S3×C12)⋊17C22, (S3×C6).22C23, D6.20(C22×S3), (C22×S3).72D6, C6.D6⋊5C22, C12⋊S3⋊20C22, C3⋊D12⋊12C22, C12.132(C22×S3), (C6×C12).162C22, (C3×C12).116C23, (C2×Dic3).117D6, (C3×Dic6)⋊27C22, (C3×Dic3).6C23, Dic3.26(C22×S3), (C6×Dic3).47C22, (S3×C2×C4)⋊4S3, (S3×C2×C12)⋊8C2, C4.63(C2×S32), (C2×C4).87S32, C3⋊1(C2×C4○D12), C32⋊4(C2×C4○D4), (C3×C6)⋊4(C4○D4), C2.11(C22×S32), C22.63(C2×S32), C3⋊1(C2×Q8⋊3S3), (C2×C6.D6)⋊3C2, (C2×C12⋊S3)⋊17C2, (C2×C3⋊D12)⋊19C2, (C2×C3⋊S3).18C23, (S3×C2×C6).105C22, (C2×C6).150(C22×S3), (C22×C3⋊S3).57C22, SmallGroup(288,949)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6.6D6
G = < a,b,c,d,e | a2=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d5 >
Subgroups: 1346 in 355 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C22×C6, C2×C4○D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, Q8⋊3S3, C2×C3⋊D4, C22×C12, C6×Q8, C6.D6, C3⋊D12, C3×Dic6, S3×C12, C6×Dic3, C6×Dic3, C12⋊S3, C6×C12, S3×C2×C6, C22×C3⋊S3, C2×C4○D12, C2×Q8⋊3S3, D6.6D6, C2×C6.D6, C2×C3⋊D12, C6×Dic6, S3×C2×C12, C2×C12⋊S3, C2×D6.6D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, C4○D12, Q8⋊3S3, S3×C23, C2×S32, C2×C4○D12, C2×Q8⋊3S3, D6.6D6, C22×S32, C2×D6.6D6
►On 48 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 37)
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 23 21 19 17 15)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 28 30 32 34 36)(37 39 41 43 45 47)(38 40 42 44 46 48)
(1 33)(2 34)(3 35)(4 36)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)
(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 34 7 28)(2 33 8 27)(3 32 9 26)(4 31 10 25)(5 30 11 36)(6 29 12 35)(13 41 19 47)(14 40 20 46)(15 39 21 45)(16 38 22 44)(17 37 23 43)(18 48 24 42)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,11,9,7,5,3)(2,12,10,8,6,4)(13,23,21,19,17,15)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,28,30,32,34,36)(37,39,41,43,45,47)(38,40,42,44,46,48), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,34,7,28)(2,33,8,27)(3,32,9,26)(4,31,10,25)(5,30,11,36)(6,29,12,35)(13,41,19,47)(14,40,20,46)(15,39,21,45)(16,38,22,44)(17,37,23,43)(18,48,24,42)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,37), (1,11,9,7,5,3)(2,12,10,8,6,4)(13,23,21,19,17,15)(14,24,22,20,18,16)(25,27,29,31,33,35)(26,28,30,32,34,36)(37,39,41,43,45,47)(38,40,42,44,46,48), (1,33)(2,34)(3,35)(4,36)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39), (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), (1,34,7,28)(2,33,8,27)(3,32,9,26)(4,31,10,25)(5,30,11,36)(6,29,12,35)(13,41,19,47)(14,40,20,46)(15,39,21,45)(16,38,22,44)(17,37,23,43)(18,48,24,42) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,37)], [(1,11,9,7,5,3),(2,12,10,8,6,4),(13,23,21,19,17,15),(14,24,22,20,18,16),(25,27,29,31,33,35),(26,28,30,32,34,36),(37,39,41,43,45,47),(38,40,42,44,46,48)], [(1,33),(2,34),(3,35),(4,36),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,37),(23,38),(24,39)], [(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)], [(1,34,7,28),(2,33,8,27),(3,32,9,26),(4,31,10,25),(5,30,11,36),(6,29,12,35),(13,41,19,47),(14,40,20,46),(15,39,21,45),(16,38,22,44),(17,37,23,43),(18,48,24,42)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 12Q | 12R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | Q8⋊3S3 | C2×S32 | C2×S32 | D6.6D6 |
| kernel | C2×D6.6D6 | D6.6D6 | C2×C6.D6 | C2×C3⋊D12 | C6×Dic6 | S3×C2×C12 | C2×C12⋊S3 | C2×Dic6 | S3×C2×C4 | Dic6 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 3 | 2 | 1 | 4 | 8 | 1 | 2 | 2 | 1 | 4 |
Matrix representation of C2×D6.6D6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 11 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 2 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,1,0,0,0,0,0,0,8,11,0,0,0,0,12,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,8,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,2,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C2×D6.6D6 in GAP, Magma, Sage, TeX
C_2\times D_6._6D_6
% in TeX
G:=Group("C2xD6.6D6"); // GroupNames label
G:=SmallGroup(288,949);
// by ID
G=gap.SmallGroup(288,949);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d^5>;
// generators/relations