direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×Dic9, D6.D9, C18.5D6, C6.5D18, (S3×C9)⋊C4, C6.5S32, C9⋊3(C4×S3), (S3×C18).C2, C2.3(S3×D9), (S3×C6).1S3, C9⋊Dic3⋊2C2, C3⋊1(C2×Dic9), (C3×C6).26D6, (C3×S3).Dic3, (C3×Dic9)⋊2C2, C3.3(S3×Dic3), (C3×C18).5C22, C32.2(C2×Dic3), (C3×C9)⋊3(C2×C4), SmallGroup(216,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — S3×Dic9 |
Generators and relations for S3×Dic9
G = < a,b,c,d | a3=b2=c18=1, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
3C2
3C2
2C3
3C22
9C4
27C4
2C6
3C6
3C6
2C9
27C2×C4
3Dic3
9Dic3
9Dic3
9C12
18Dic3
2C18
3C18
3C18
3Dic9
6Dic9
Smallest permutation representation of S3×Dic9
►On 72 points
Generators in S72
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)
(1 52)(2 53)(3 54)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 71)(20 72)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)
(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 67 68 69 70 71 72)
(1 29 10 20)(2 28 11 19)(3 27 12 36)(4 26 13 35)(5 25 14 34)(6 24 15 33)(7 23 16 32)(8 22 17 31)(9 21 18 30)(37 60 46 69)(38 59 47 68)(39 58 48 67)(40 57 49 66)(41 56 50 65)(42 55 51 64)(43 72 52 63)(44 71 53 62)(45 70 54 61)
G:=sub<Sym(72)| (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (1,52)(2,53)(3,54)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,71)(20,72)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,29,10,20)(2,28,11,19)(3,27,12,36)(4,26,13,35)(5,25,14,34)(6,24,15,33)(7,23,16,32)(8,22,17,31)(9,21,18,30)(37,60,46,69)(38,59,47,68)(39,58,48,67)(40,57,49,66)(41,56,50,65)(42,55,51,64)(43,72,52,63)(44,71,53,62)(45,70,54,61)>;
G:=Group( (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (1,52)(2,53)(3,54)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,43)(11,44)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,71)(20,72)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70), (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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,29,10,20)(2,28,11,19)(3,27,12,36)(4,26,13,35)(5,25,14,34)(6,24,15,33)(7,23,16,32)(8,22,17,31)(9,21,18,30)(37,60,46,69)(38,59,47,68)(39,58,48,67)(40,57,49,66)(41,56,50,65)(42,55,51,64)(43,72,52,63)(44,71,53,62)(45,70,54,61) );
G=PermutationGroup([[(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72)], [(1,52),(2,53),(3,54),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,43),(11,44),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,71),(20,72),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70)], [(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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,29,10,20),(2,28,11,19),(3,27,12,36),(4,26,13,35),(5,25,14,34),(6,24,15,33),(7,23,16,32),(8,22,17,31),(9,21,18,30),(37,60,46,69),(38,59,47,68),(39,58,48,67),(40,57,49,66),(41,56,50,65),(42,55,51,64),(43,72,52,63),(44,71,53,62),(45,70,54,61)]])
S3×Dic9 is a maximal subgroup of
D12⋊5D9 D12⋊D9 C4×S3×D9 Dic3.D18 D18.4D6
S3×Dic9 is a maximal quotient of D6.Dic9 Dic3⋊Dic9 D6⋊Dic9
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 9 | 9 | 27 | 27 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | Dic3 | D6 | D9 | C4×S3 | Dic9 | D18 | S32 | S3×Dic3 | S3×D9 | S3×Dic9 |
| kernel | S3×Dic9 | C3×Dic9 | C9⋊Dic3 | S3×C18 | S3×C9 | Dic9 | S3×C6 | C18 | C3×S3 | C3×C6 | D6 | C9 | S3 | C6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 6 | 3 | 1 | 1 | 3 | 3 |
Matrix representation of S3×Dic9 ►in GL4(𝔽37) generated by
| 36 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 36 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 17 | 26 |
| 0 | 0 | 11 | 6 |
| 31 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [36,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,36,36,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,17,11,0,0,26,6],[31,0,0,0,0,31,0,0,0,0,0,36,0,0,36,0] >;
S3×Dic9 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_9 % in TeX
G:=Group("S3xDic9"); // GroupNames label
G:=SmallGroup(216,30);
// by ID
G=gap.SmallGroup(216,30);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,1065,453,1444,2603]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^18=1,d^2=c^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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