direct product, metacyclic, supersoluble, monomial
Aliases: C9×C4.Dic3, C12.1C36, C36.79D6, C36.9Dic3, C62.18C12, C3⋊C8⋊5C18, C4.(C9×Dic3), (C2×C6).6C36, (C3×C36).9C4, C6.6(C2×C36), (C6×C18).2C4, (C3×C9)⋊7M4(2), C4.15(S3×C18), (C6×C36).17C2, (C6×C12).31C6, (C2×C12).5C18, (C2×C36).18S3, C3⋊2(C9×M4(2)), C12.119(S3×C6), C12.15(C2×C18), (C3×C12).16C12, C22.(C9×Dic3), C6.30(C6×Dic3), C2.3(Dic3×C18), (C2×C18).2Dic3, (C3×C36).53C22, C12.19(C3×Dic3), C18.18(C2×Dic3), C32.3(C3×M4(2)), (C9×C3⋊C8)⋊12C2, (C3×C3⋊C8).8C6, (C2×C4).2(S3×C9), (C3×C4.Dic3).C3, (C3×C12).90(C2×C6), (C2×C12).40(C3×S3), (C3×C6).52(C2×C12), (C3×C18).28(C2×C4), (C2×C6).9(C3×Dic3), C3.4(C3×C4.Dic3), SmallGroup(432,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×C4.Dic3
G = < a,b,c,d | a9=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 116 in 76 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, M4(2), C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C4.Dic3, C3×M4(2), C3×C18, C3×C18, C72, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C9×M4(2), C3×C4.Dic3, C9×C3⋊C8, C6×C36, C9×C4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C9, Dic3, C12, D6, C2×C6, M4(2), C18, C3×S3, C2×Dic3, C2×C12, C36, C2×C18, C3×Dic3, S3×C6, C4.Dic3, C3×M4(2), S3×C9, C2×C36, C6×Dic3, C9×Dic3, S3×C18, C9×M4(2), C3×C4.Dic3, Dic3×C18, C9×C4.Dic3
►On 72 points
(1 33 15 9 29 23 5 25 19)(2 34 16 10 30 24 6 26 20)(3 35 17 11 31 13 7 27 21)(4 36 18 12 32 14 8 28 22)(37 72 55 41 64 59 45 68 51)(38 61 56 42 65 60 46 69 52)(39 62 57 43 66 49 47 70 53)(40 63 58 44 67 50 48 71 54)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 52 55 58)(50 53 56 59)(51 54 57 60)(61 64 67 70)(62 65 68 71)(63 66 69 72)
(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 39 10 48 7 45 4 42)(2 44 11 41 8 38 5 47)(3 37 12 46 9 43 6 40)(13 59 22 56 19 53 16 50)(14 52 23 49 20 58 17 55)(15 57 24 54 21 51 18 60)(25 70 34 67 31 64 28 61)(26 63 35 72 32 69 29 66)(27 68 36 65 33 62 30 71)
G:=sub<Sym(72)| (1,33,15,9,29,23,5,25,19)(2,34,16,10,30,24,6,26,20)(3,35,17,11,31,13,7,27,21)(4,36,18,12,32,14,8,28,22)(37,72,55,41,64,59,45,68,51)(38,61,56,42,65,60,46,69,52)(39,62,57,43,66,49,47,70,53)(40,63,58,44,67,50,48,71,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72), (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), (1,39,10,48,7,45,4,42)(2,44,11,41,8,38,5,47)(3,37,12,46,9,43,6,40)(13,59,22,56,19,53,16,50)(14,52,23,49,20,58,17,55)(15,57,24,54,21,51,18,60)(25,70,34,67,31,64,28,61)(26,63,35,72,32,69,29,66)(27,68,36,65,33,62,30,71)>;
G:=Group( (1,33,15,9,29,23,5,25,19)(2,34,16,10,30,24,6,26,20)(3,35,17,11,31,13,7,27,21)(4,36,18,12,32,14,8,28,22)(37,72,55,41,64,59,45,68,51)(38,61,56,42,65,60,46,69,52)(39,62,57,43,66,49,47,70,53)(40,63,58,44,67,50,48,71,54), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,22,19,16)(14,23,20,17)(15,24,21,18)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,52,55,58)(50,53,56,59)(51,54,57,60)(61,64,67,70)(62,65,68,71)(63,66,69,72), (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), (1,39,10,48,7,45,4,42)(2,44,11,41,8,38,5,47)(3,37,12,46,9,43,6,40)(13,59,22,56,19,53,16,50)(14,52,23,49,20,58,17,55)(15,57,24,54,21,51,18,60)(25,70,34,67,31,64,28,61)(26,63,35,72,32,69,29,66)(27,68,36,65,33,62,30,71) );
G=PermutationGroup([[(1,33,15,9,29,23,5,25,19),(2,34,16,10,30,24,6,26,20),(3,35,17,11,31,13,7,27,21),(4,36,18,12,32,14,8,28,22),(37,72,55,41,64,59,45,68,51),(38,61,56,42,65,60,46,69,52),(39,62,57,43,66,49,47,70,53),(40,63,58,44,67,50,48,71,54)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,22,19,16),(14,23,20,17),(15,24,21,18),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,52,55,58),(50,53,56,59),(51,54,57,60),(61,64,67,70),(62,65,68,71),(63,66,69,72)], [(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)], [(1,39,10,48,7,45,4,42),(2,44,11,41,8,38,5,47),(3,37,12,46,9,43,6,40),(13,59,22,56,19,53,16,50),(14,52,23,49,20,58,17,55),(15,57,24,54,21,51,18,60),(25,70,34,67,31,64,28,61),(26,63,35,72,32,69,29,66),(27,68,36,65,33,62,30,71)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6M | 8A | 8B | 8C | 8D | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 18A | ··· | 18F | 18G | ··· | 18AD | 24A | ··· | 24H | 36A | ··· | 36L | 36M | ··· | 36AP | 72A | ··· | 72X |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C9 | C12 | C12 | C18 | C18 | C36 | C36 | S3 | Dic3 | D6 | Dic3 | M4(2) | C3×S3 | C3×Dic3 | S3×C6 | C3×Dic3 | C4.Dic3 | C3×M4(2) | S3×C9 | C9×Dic3 | S3×C18 | C9×Dic3 | C9×M4(2) | C3×C4.Dic3 | C9×C4.Dic3 |
| kernel | C9×C4.Dic3 | C9×C3⋊C8 | C6×C36 | C3×C4.Dic3 | C3×C36 | C6×C18 | C3×C3⋊C8 | C6×C12 | C4.Dic3 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C12 | C2×C6 | C2×C36 | C36 | C36 | C2×C18 | C3×C9 | C2×C12 | C12 | C12 | C2×C6 | C9 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 4 | 4 | 12 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 8 | 24 |
Matrix representation of C9×C4.Dic3 ►in GL2(𝔽37) generated by
| 9 | 0 |
| 0 | 9 |
| 31 | 0 |
| 0 | 6 |
| 8 | 0 |
| 0 | 23 |
| 0 | 19 |
| 12 | 0 |
G:=sub<GL(2,GF(37))| [9,0,0,9],[31,0,0,6],[8,0,0,23],[0,12,19,0] >;
C9×C4.Dic3 in GAP, Magma, Sage, TeX
C_9\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C9xC4.Dic3"); // GroupNames label
G:=SmallGroup(432,127);
// by ID
G=gap.SmallGroup(432,127);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,84,1037,142,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations