direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.59D6, C62.155D6, (C6×C12)⋊13C6, (C6×C12)⋊14S3, C12⋊S3⋊14C6, C32⋊7D4⋊9C6, C12.105(S3×C6), (C3×C12).211D6, C33⋊28(C4○D4), C62.78(C2×C6), C32⋊4Q8⋊14C6, C32⋊26(C4○D12), (C3×C62).64C22, (C32×C6).88C23, (C32×C12).100C22, (C3×C6×C12)⋊8C2, C6.55(S3×C2×C6), (C4×C3⋊S3)⋊13C6, (C2×C12)⋊4(C3×S3), C4.16(C6×C3⋊S3), C3⋊5(C3×C4○D12), (C12×C3⋊S3)⋊17C2, (C2×C12)⋊7(C3⋊S3), C12.98(C2×C3⋊S3), (C2×C6).78(S3×C6), C22.2(C6×C3⋊S3), (C3×C12).76(C2×C6), (C3×C12⋊S3)⋊17C2, C32⋊12(C3×C4○D4), C6.55(C22×C3⋊S3), (C6×C3⋊S3).61C22, (C3×C32⋊7D4)⋊11C2, C3⋊Dic3.22(C2×C6), (C3×C6).62(C22×C6), (C3×C32⋊4Q8)⋊17C2, (C3×C6).177(C22×S3), (C3×C3⋊Dic3).59C22, C2.5(C2×C6×C3⋊S3), (C2×C4)⋊3(C3×C3⋊S3), (C2×C6).28(C2×C3⋊S3), (C2×C3⋊S3).22(C2×C6), SmallGroup(432,713)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.59D6
G = < a,b,c,d | a3=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 884 in 304 conjugacy classes, 94 normal (30 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, C62, C4○D12, C3×C4○D4, C3×C3⋊S3, C32×C6, C32×C6, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C62, C3×C4○D12, C12.59D6, C3×C32⋊4Q8, C12×C3⋊S3, C3×C12⋊S3, C3×C32⋊7D4, C3×C6×C12, C3×C12.59D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, C4○D12, C3×C4○D4, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×C4○D12, C12.59D6, C2×C6×C3⋊S3, C3×C12.59D6
►On 72 points
(1 24 29)(2 13 30)(3 14 31)(4 15 32)(5 16 33)(6 17 34)(7 18 35)(8 19 36)(9 20 25)(10 21 26)(11 22 27)(12 23 28)(37 52 71)(38 53 72)(39 54 61)(40 55 62)(41 56 63)(42 57 64)(43 58 65)(44 59 66)(45 60 67)(46 49 68)(47 50 69)(48 51 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 24 29)(2 13 30)(3 14 31)(4 15 32)(5 16 33)(6 17 34)(7 18 35)(8 19 36)(9 20 25)(10 21 26)(11 22 27)(12 23 28)(37 65 52 43 71 58)(38 66 53 44 72 59)(39 67 54 45 61 60)(40 68 55 46 62 49)(41 69 56 47 63 50)(42 70 57 48 64 51)
(1 47 7 41)(2 40 8 46)(3 45 9 39)(4 38 10 44)(5 43 11 37)(6 48 12 42)(13 55 19 49)(14 60 20 54)(15 53 21 59)(16 58 22 52)(17 51 23 57)(18 56 24 50)(25 61 31 67)(26 66 32 72)(27 71 33 65)(28 64 34 70)(29 69 35 63)(30 62 36 68)
G:=sub<Sym(72)| (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,52,71)(38,53,72)(39,54,61)(40,55,62)(41,56,63)(42,57,64)(43,58,65)(44,59,66)(45,60,67)(46,49,68)(47,50,69)(48,51,70), (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), (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,65,52,43,71,58)(38,66,53,44,72,59)(39,67,54,45,61,60)(40,68,55,46,62,49)(41,69,56,47,63,50)(42,70,57,48,64,51), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,61,31,67)(26,66,32,72)(27,71,33,65)(28,64,34,70)(29,69,35,63)(30,62,36,68)>;
G:=Group( (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,52,71)(38,53,72)(39,54,61)(40,55,62)(41,56,63)(42,57,64)(43,58,65)(44,59,66)(45,60,67)(46,49,68)(47,50,69)(48,51,70), (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), (1,24,29)(2,13,30)(3,14,31)(4,15,32)(5,16,33)(6,17,34)(7,18,35)(8,19,36)(9,20,25)(10,21,26)(11,22,27)(12,23,28)(37,65,52,43,71,58)(38,66,53,44,72,59)(39,67,54,45,61,60)(40,68,55,46,62,49)(41,69,56,47,63,50)(42,70,57,48,64,51), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,61,31,67)(26,66,32,72)(27,71,33,65)(28,64,34,70)(29,69,35,63)(30,62,36,68) );
G=PermutationGroup([[(1,24,29),(2,13,30),(3,14,31),(4,15,32),(5,16,33),(6,17,34),(7,18,35),(8,19,36),(9,20,25),(10,21,26),(11,22,27),(12,23,28),(37,52,71),(38,53,72),(39,54,61),(40,55,62),(41,56,63),(42,57,64),(43,58,65),(44,59,66),(45,60,67),(46,49,68),(47,50,69),(48,51,70)], [(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)], [(1,24,29),(2,13,30),(3,14,31),(4,15,32),(5,16,33),(6,17,34),(7,18,35),(8,19,36),(9,20,25),(10,21,26),(11,22,27),(12,23,28),(37,65,52,43,71,58),(38,66,53,44,72,59),(39,67,54,45,61,60),(40,68,55,46,62,49),(41,69,56,47,63,50),(42,70,57,48,64,51)], [(1,47,7,41),(2,40,8,46),(3,45,9,39),(4,38,10,44),(5,43,11,37),(6,48,12,42),(13,55,19,49),(14,60,20,54),(15,53,21,59),(16,58,22,52),(17,51,23,57),(18,56,24,50),(25,61,31,67),(26,66,32,72),(27,71,33,65),(28,64,34,70),(29,69,35,63),(30,62,36,68)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6AN | 6AO | 6AP | 6AQ | 6AR | 12A | 12B | 12C | 12D | 12E | ··· | 12BB | 12BC | 12BD | 12BE | 12BF |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D6 | D6 | C4○D4 | C3×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C4○D12 |
| kernel | C3×C12.59D6 | C3×C32⋊4Q8 | C12×C3⋊S3 | C3×C12⋊S3 | C3×C32⋊7D4 | C3×C6×C12 | C12.59D6 | C32⋊4Q8 | C4×C3⋊S3 | C12⋊S3 | C32⋊7D4 | C6×C12 | C6×C12 | C3×C12 | C62 | C33 | C2×C12 | C12 | C2×C6 | C32 | C32 | C3 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 8 | 4 | 2 | 8 | 16 | 8 | 16 | 4 | 32 |
Matrix representation of C3×C12.59D6 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 12 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[0,9,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
C3×C12.59D6 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{59}D_6 % in TeX
G:=Group("C3xC12.59D6"); // GroupNames label
G:=SmallGroup(432,713);
// by ID
G=gap.SmallGroup(432,713);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations