direct product, metabelian, supersoluble, monomial
Aliases: S3×D36, C36⋊4D6, D18⋊1D6, C12⋊1D18, Dic3⋊3D18, D6.10D18, C9⋊1(S3×D4), C4⋊2(S3×D9), C12.51S32, (S3×C9)⋊1D4, (C4×S3)⋊3D9, C3⋊1(C2×D36), (C3×S3).D12, (S3×C36)⋊1C2, (C3×D36)⋊4C2, C3.1(S3×D12), C36⋊S3⋊4C2, C3⋊D36⋊1C2, (S3×C12).7S3, (C3×C36)⋊1C22, (S3×C6).30D6, (C3×C12).98D6, (C6×D9)⋊1C22, C32.2(C2×D12), C6.12(C22×D9), C18.12(C22×S3), (C3×C18).12C23, (C3×Dic3).30D6, (C9×Dic3)⋊3C22, (S3×C18).13C22, (C2×S3×D9)⋊1C2, (C3×C9)⋊1(C2×D4), C6.31(C2×S32), C2.15(C2×S3×D9), (C2×C9⋊S3)⋊1C22, (C3×C6).80(C22×S3), SmallGroup(432,291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×D36
G = < a,b,c,d | a3=b2=c36=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1480 in 178 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C2×C4, D4, C23, C9, C9, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, D9, C18, C18, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×S3, C3×C9, C36, C36, D18, D18, C2×C18, C3×Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×D12, S3×D4, C3×D9, S3×C9, C9⋊S3, C3×C18, D36, D36, C2×C36, C22×D9, C3⋊D12, S3×C12, C3×D12, C12⋊S3, C2×S32, C9×Dic3, C3×C36, S3×D9, C6×D9, S3×C18, C2×C9⋊S3, C2×D36, S3×D12, C3⋊D36, C3×D36, S3×C36, C36⋊S3, C2×S3×D9, S3×D36
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, D18, S32, C2×D12, S3×D4, D36, C22×D9, C2×S32, S3×D9, C2×D36, S3×D12, C2×S3×D9, S3×D36
►On 72 points
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 37)(33 38)(34 39)(35 40)(36 41)
(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 41)(2 40)(3 39)(4 38)(5 37)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)
G:=sub<Sym(72)| (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,37)(33,38)(34,39)(35,40)(36,41), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)>;
G:=Group( (1,13,25)(2,14,26)(3,15,27)(4,16,28)(5,17,29)(6,18,30)(7,19,31)(8,20,32)(9,21,33)(10,22,34)(11,23,35)(12,24,36)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,37)(33,38)(34,39)(35,40)(36,41), (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), (1,41)(2,40)(3,39)(4,38)(5,37)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42) );
G=PermutationGroup([[(1,13,25),(2,14,26),(3,15,27),(4,16,28),(5,17,29),(6,18,30),(7,19,31),(8,20,32),(9,21,33),(10,22,34),(11,23,35),(12,24,36),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,37),(33,38),(34,39),(35,40),(36,41)], [(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)], [(1,41),(2,40),(3,39),(4,38),(5,37),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 18D | 18E | 18F | 18G | ··· | 18L | 36A | ··· | 36F | 36G | ··· | 36L | 36M | ··· | 36R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 3 | 3 | 18 | 18 | 54 | 54 | 2 | 2 | 4 | 2 | 6 | 2 | 2 | 4 | 6 | 6 | 36 | 36 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D6 | D6 | D6 | D6 | D6 | D9 | D12 | D18 | D18 | D18 | D36 | S32 | S3×D4 | C2×S32 | S3×D9 | S3×D12 | C2×S3×D9 | S3×D36 |
| kernel | S3×D36 | C3⋊D36 | C3×D36 | S3×C36 | C36⋊S3 | C2×S3×D9 | D36 | S3×C12 | S3×C9 | C36 | D18 | C3×Dic3 | C3×C12 | S3×C6 | C4×S3 | C3×S3 | Dic3 | C12 | D6 | S3 | C12 | C9 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 12 | 1 | 1 | 1 | 3 | 2 | 3 | 6 |
Matrix representation of S3×D36 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 34 | 0 | 0 | 0 | 0 |
| 26 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 10 |
| 0 | 0 | 0 | 0 | 34 | 31 |
| 22 | 3 | 0 | 0 | 0 | 0 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 34 |
| 0 | 0 | 0 | 0 | 22 | 20 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,26,0,0,0,0,34,22,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,11,34,0,0,0,0,10,31],[22,24,0,0,0,0,3,15,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,17,22,0,0,0,0,34,20] >;
S3×D36 in GAP, Magma, Sage, TeX
S_3\times D_{36} % in TeX
G:=Group("S3xD36"); // GroupNames label
G:=SmallGroup(432,291);
// by ID
G=gap.SmallGroup(432,291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^36=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations