direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×S3×D11, D33⋊C6, C33⋊4D6, C32⋊3D22, C33⋊(C2×C6), (S3×C11)⋊C6, C11⋊1(S3×C6), (C3×D11)⋊C6, C3⋊1(C6×D11), (S3×C33)⋊2C2, (C3×D33)⋊1C2, (C3×C33)⋊1C22, (C32×D11)⋊1C2, SmallGroup(396,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C3×S3×D11 |
Generators and relations for C3×S3×D11
G = < a,b,c,d,e | a3=b3=c2=d11=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
3C2
11C2
33C2
2C3
33C22
3C6
11C6
11C6
11S3
22C6
33C6
3C22
3D11
2C33
11D6
33C2×C6
11C3×S3
11C3×C6
3D22
3C66
11S3×C6
Smallest permutation representation of C3×S3×D11
►On 66 points
Generators in S66
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(34 45 56)(35 46 57)(36 47 58)(37 48 59)(38 49 60)(39 50 61)(40 51 62)(41 52 63)(42 53 64)(43 54 65)(44 55 66)
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)(34 56 45)(35 57 46)(36 58 47)(37 59 48)(38 60 49)(39 61 50)(40 62 51)(41 63 52)(42 64 53)(43 65 54)(44 66 55)
(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)
(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 49 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)
G:=sub<Sym(66)| (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55), (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)>;
G:=Group( (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,45,56)(35,46,57)(36,47,58)(37,48,59)(38,49,60)(39,50,61)(40,51,62)(41,52,63)(42,53,64)(43,54,65)(44,55,66), (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31)(34,56,45)(35,57,46)(36,58,47)(37,59,48)(38,60,49)(39,61,50)(40,62,51)(41,63,52)(42,64,53)(43,65,54)(44,66,55), (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66), (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65) );
G=PermutationGroup([[(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31),(34,45,56),(35,46,57),(36,47,58),(37,48,59),(38,49,60),(39,50,61),(40,51,62),(41,52,63),(42,53,64),(43,54,65),(44,55,66)], [(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31),(34,56,45),(35,57,46),(36,58,47),(37,59,48),(38,60,49),(39,61,50),(40,62,51),(41,63,52),(42,64,53),(43,65,54),(44,66,55)], [(1,43),(2,44),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,66)], [(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),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33J | 33K | ··· | 33Y | 66A | ··· | 66J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 3 | 11 | 33 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 11 | 11 | 22 | 22 | 22 | 33 | 33 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D6 | C3×S3 | D11 | S3×C6 | D22 | C3×D11 | C6×D11 | S3×D11 | C3×S3×D11 |
| kernel | C3×S3×D11 | C32×D11 | S3×C33 | C3×D33 | S3×D11 | S3×C11 | C3×D11 | D33 | C3×D11 | C33 | D11 | C3×S3 | C11 | C32 | S3 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 5 | 2 | 5 | 10 | 10 | 5 | 10 |
Matrix representation of C3×S3×D11 ►in GL4(𝔽67) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 37 | 0 | 0 | 0 |
| 0 | 29 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 48 | 1 |
| 0 | 0 | 55 | 57 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 66 |
| 0 | 0 | 32 | 10 |
G:=sub<GL(4,GF(67))| [37,0,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[37,0,0,0,0,29,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,48,55,0,0,1,57],[1,0,0,0,0,1,0,0,0,0,57,32,0,0,66,10] >;
C3×S3×D11 in GAP, Magma, Sage, TeX
C_3\times S_3\times D_{11} % in TeX
G:=Group("C3xS3xD11"); // GroupNames label
G:=SmallGroup(396,19);
// by ID
G=gap.SmallGroup(396,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-11,248,9004]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^11=e^2=1,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,c*e=e*c,e*d*e=d^-1>;
// generators/relations
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