direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.58D6, C62.24C12, C33⋊17M4(2), C62.24Dic3, (C6×C12).30C6, (C6×C12).40S3, (C3×C62).9C4, C12.104(S3×C6), (C3×C12).15C12, C32⋊4C8⋊16C6, (C3×C12).226D6, (C32×C12).9C4, C12.7(C3×Dic3), C6.25(C6×Dic3), (C3×C12).20Dic3, C12.11(C3⋊Dic3), C32⋊11(C3×M4(2)), (C32×C12).94C22, C32⋊10(C4.Dic3), (C3×C6×C12).8C2, C4.(C3×C3⋊Dic3), C4.15(C6×C3⋊S3), C12.97(C2×C3⋊S3), C2.3(C6×C3⋊Dic3), C3⋊2(C3×C4.Dic3), (C2×C12).28(C3×S3), (C3×C12).99(C2×C6), (C3×C6).62(C2×C12), C22.(C3×C3⋊Dic3), C6.21(C2×C3⋊Dic3), (C2×C12).25(C3⋊S3), (C3×C32⋊4C8)⋊20C2, (C32×C6).69(C2×C4), (C2×C6).6(C3⋊Dic3), (C2×C6).25(C3×Dic3), (C3×C6).67(C2×Dic3), (C2×C4).2(C3×C3⋊S3), SmallGroup(432,486)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.58D6
G = < a,b,c,d | a3=b12=c6=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >
Subgroups: 356 in 184 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, M4(2), C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C33, C3×C12, C3×C12, C3×C12, C62, C62, C62, C4.Dic3, C3×M4(2), C32×C6, C32×C6, C3×C3⋊C8, C32⋊4C8, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C3×C4.Dic3, C12.58D6, C3×C32⋊4C8, C3×C6×C12, C3×C12.58D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C4.Dic3, C3×M4(2), C3×C3⋊S3, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C3×C4.Dic3, C12.58D6, C6×C3⋊Dic3, C3×C12.58D6
►On 72 points
(1 71 52)(2 72 53)(3 61 54)(4 62 55)(5 63 56)(6 64 57)(7 65 58)(8 66 59)(9 67 60)(10 68 49)(11 69 50)(12 70 51)(13 28 43)(14 29 44)(15 30 45)(16 31 46)(17 32 47)(18 33 48)(19 34 37)(20 35 38)(21 36 39)(22 25 40)(23 26 41)(24 27 42)
(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 71 52)(2 72 53)(3 61 54)(4 62 55)(5 63 56)(6 64 57)(7 65 58)(8 66 59)(9 67 60)(10 68 49)(11 69 50)(12 70 51)(13 37 28 19 43 34)(14 38 29 20 44 35)(15 39 30 21 45 36)(16 40 31 22 46 25)(17 41 32 23 47 26)(18 42 33 24 48 27)
(1 44 4 47 7 38 10 41)(2 37 5 40 8 43 11 46)(3 42 6 45 9 48 12 39)(13 69 16 72 19 63 22 66)(14 62 17 65 20 68 23 71)(15 67 18 70 21 61 24 64)(25 59 28 50 31 53 34 56)(26 52 29 55 32 58 35 49)(27 57 30 60 33 51 36 54)
G:=sub<Sym(72)| (1,71,52)(2,72,53)(3,61,54)(4,62,55)(5,63,56)(6,64,57)(7,65,58)(8,66,59)(9,67,60)(10,68,49)(11,69,50)(12,70,51)(13,28,43)(14,29,44)(15,30,45)(16,31,46)(17,32,47)(18,33,48)(19,34,37)(20,35,38)(21,36,39)(22,25,40)(23,26,41)(24,27,42), (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), (1,71,52)(2,72,53)(3,61,54)(4,62,55)(5,63,56)(6,64,57)(7,65,58)(8,66,59)(9,67,60)(10,68,49)(11,69,50)(12,70,51)(13,37,28,19,43,34)(14,38,29,20,44,35)(15,39,30,21,45,36)(16,40,31,22,46,25)(17,41,32,23,47,26)(18,42,33,24,48,27), (1,44,4,47,7,38,10,41)(2,37,5,40,8,43,11,46)(3,42,6,45,9,48,12,39)(13,69,16,72,19,63,22,66)(14,62,17,65,20,68,23,71)(15,67,18,70,21,61,24,64)(25,59,28,50,31,53,34,56)(26,52,29,55,32,58,35,49)(27,57,30,60,33,51,36,54)>;
G:=Group( (1,71,52)(2,72,53)(3,61,54)(4,62,55)(5,63,56)(6,64,57)(7,65,58)(8,66,59)(9,67,60)(10,68,49)(11,69,50)(12,70,51)(13,28,43)(14,29,44)(15,30,45)(16,31,46)(17,32,47)(18,33,48)(19,34,37)(20,35,38)(21,36,39)(22,25,40)(23,26,41)(24,27,42), (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), (1,71,52)(2,72,53)(3,61,54)(4,62,55)(5,63,56)(6,64,57)(7,65,58)(8,66,59)(9,67,60)(10,68,49)(11,69,50)(12,70,51)(13,37,28,19,43,34)(14,38,29,20,44,35)(15,39,30,21,45,36)(16,40,31,22,46,25)(17,41,32,23,47,26)(18,42,33,24,48,27), (1,44,4,47,7,38,10,41)(2,37,5,40,8,43,11,46)(3,42,6,45,9,48,12,39)(13,69,16,72,19,63,22,66)(14,62,17,65,20,68,23,71)(15,67,18,70,21,61,24,64)(25,59,28,50,31,53,34,56)(26,52,29,55,32,58,35,49)(27,57,30,60,33,51,36,54) );
G=PermutationGroup([[(1,71,52),(2,72,53),(3,61,54),(4,62,55),(5,63,56),(6,64,57),(7,65,58),(8,66,59),(9,67,60),(10,68,49),(11,69,50),(12,70,51),(13,28,43),(14,29,44),(15,30,45),(16,31,46),(17,32,47),(18,33,48),(19,34,37),(20,35,38),(21,36,39),(22,25,40),(23,26,41),(24,27,42)], [(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)], [(1,71,52),(2,72,53),(3,61,54),(4,62,55),(5,63,56),(6,64,57),(7,65,58),(8,66,59),(9,67,60),(10,68,49),(11,69,50),(12,70,51),(13,37,28,19,43,34),(14,38,29,20,44,35),(15,39,30,21,45,36),(16,40,31,22,46,25),(17,41,32,23,47,26),(18,42,33,24,48,27)], [(1,44,4,47,7,38,10,41),(2,37,5,40,8,43,11,46),(3,42,6,45,9,48,12,39),(13,69,16,72,19,63,22,66),(14,62,17,65,20,68,23,71),(15,67,18,70,21,61,24,64),(25,59,28,50,31,53,34,56),(26,52,29,55,32,58,35,49),(27,57,30,60,33,51,36,54)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6AN | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12BB | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 18 | ··· | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | Dic3 | M4(2) | C3×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | C4.Dic3 | C3×M4(2) | C3×C4.Dic3 |
| kernel | C3×C12.58D6 | C3×C32⋊4C8 | C3×C6×C12 | C12.58D6 | C32×C12 | C3×C62 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C6×C12 | C3×C12 | C3×C12 | C62 | C33 | C2×C12 | C12 | C12 | C2×C6 | C32 | C32 | C3 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 8 | 8 | 16 | 4 | 32 |
Matrix representation of C3×C12.58D6 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 65 | 0 | 0 | 0 |
| 20 | 9 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 8 | 0 | 0 | 0 |
| 53 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 9 |
| 20 | 17 | 0 | 0 |
| 58 | 53 | 0 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[65,20,0,0,0,9,0,0,0,0,27,0,0,0,0,27],[8,53,0,0,0,64,0,0,0,0,8,0,0,0,0,9],[20,58,0,0,17,53,0,0,0,0,0,3,0,0,64,0] >;
C3×C12.58D6 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{58}D_6 % in TeX
G:=Group("C3xC12.58D6"); // GroupNames label
G:=SmallGroup(432,486);
// by ID
G=gap.SmallGroup(432,486);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^6=1,d^2=b^3,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