direct product, metabelian, supersoluble, monomial
Aliases: C32×C4.Dic3, C62.22C12, C12.15C62, C33⋊16M4(2), C62.14Dic3, C6.7(C6×C12), (C6×C12).56S3, C12.1(C3×C12), (C6×C12).33C6, (C3×C62).7C4, C12.124(S3×C6), (C3×C12).18C12, C4.(C32×Dic3), (C3×C12).238D6, (C32×C12).8C4, C6.36(C6×Dic3), (C3×C12).31Dic3, C12.21(C3×Dic3), C3⋊2(C32×M4(2)), C22.(C32×Dic3), C32⋊10(C3×M4(2)), (C32×C12).90C22, C3⋊C8⋊5(C3×C6), (C3×C3⋊C8)⋊12C6, C4.15(S3×C3×C6), (C3×C6×C12).7C2, C2.3(Dic3×C3×C6), (C32×C3⋊C8)⋊19C2, (C3×C6).57(C2×C12), (C2×C12).45(C3×S3), (C2×C6).14(C3×C12), (C3×C12).95(C2×C6), (C2×C12).10(C3×C6), (C2×C4).2(S3×C32), (C32×C6).65(C2×C4), (C3×C6).77(C2×Dic3), (C2×C6).11(C3×Dic3), SmallGroup(432,470)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4.Dic3
G = < a,b,c,d,e | a3=b3=c4=1, d6=c2, e2=c2d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >
Subgroups: 296 in 184 conjugacy classes, 90 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, C3×C24, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C3×C4.Dic3, C32×M4(2), C32×C3⋊C8, C3×C6×C12, C32×C4.Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C32, Dic3, C12, D6, C2×C6, M4(2), C3×S3, C3×C6, C2×Dic3, C2×C12, C3×Dic3, C3×C12, S3×C6, C62, C4.Dic3, C3×M4(2), S3×C32, C6×Dic3, C6×C12, C32×Dic3, S3×C3×C6, C3×C4.Dic3, C32×M4(2), Dic3×C3×C6, C32×C4.Dic3
►On 72 points
(1 33 23)(2 34 24)(3 35 13)(4 36 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)(11 31 21)(12 32 22)(37 66 51)(38 67 52)(39 68 53)(40 69 54)(41 70 55)(42 71 56)(43 72 57)(44 61 58)(45 62 59)(46 63 60)(47 64 49)(48 65 50)
(1 25 19)(2 26 20)(3 27 21)(4 28 22)(5 29 23)(6 30 24)(7 31 13)(8 32 14)(9 33 15)(10 34 16)(11 35 17)(12 36 18)(37 62 55)(38 63 56)(39 64 57)(40 65 58)(41 66 59)(42 67 60)(43 68 49)(44 69 50)(45 70 51)(46 71 52)(47 72 53)(48 61 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 48 10 45 7 42 4 39)(2 41 11 38 8 47 5 44)(3 46 12 43 9 40 6 37)(13 60 22 57 19 54 16 51)(14 53 23 50 20 59 17 56)(15 58 24 55 21 52 18 49)(25 61 34 70 31 67 28 64)(26 66 35 63 32 72 29 69)(27 71 36 68 33 65 30 62)
G:=sub<Sym(72)| (1,33,23)(2,34,24)(3,35,13)(4,36,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(11,31,21)(12,32,22)(37,66,51)(38,67,52)(39,68,53)(40,69,54)(41,70,55)(42,71,56)(43,72,57)(44,61,58)(45,62,59)(46,63,60)(47,64,49)(48,65,50), (1,25,19)(2,26,20)(3,27,21)(4,28,22)(5,29,23)(6,30,24)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18)(37,62,55)(38,63,56)(39,64,57)(40,65,58)(41,66,59)(42,67,60)(43,68,49)(44,69,50)(45,70,51)(46,71,52)(47,72,53)(48,61,54), (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), (1,48,10,45,7,42,4,39)(2,41,11,38,8,47,5,44)(3,46,12,43,9,40,6,37)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,61,34,70,31,67,28,64)(26,66,35,63,32,72,29,69)(27,71,36,68,33,65,30,62)>;
G:=Group( (1,33,23)(2,34,24)(3,35,13)(4,36,14)(5,25,15)(6,26,16)(7,27,17)(8,28,18)(9,29,19)(10,30,20)(11,31,21)(12,32,22)(37,66,51)(38,67,52)(39,68,53)(40,69,54)(41,70,55)(42,71,56)(43,72,57)(44,61,58)(45,62,59)(46,63,60)(47,64,49)(48,65,50), (1,25,19)(2,26,20)(3,27,21)(4,28,22)(5,29,23)(6,30,24)(7,31,13)(8,32,14)(9,33,15)(10,34,16)(11,35,17)(12,36,18)(37,62,55)(38,63,56)(39,64,57)(40,65,58)(41,66,59)(42,67,60)(43,68,49)(44,69,50)(45,70,51)(46,71,52)(47,72,53)(48,61,54), (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), (1,48,10,45,7,42,4,39)(2,41,11,38,8,47,5,44)(3,46,12,43,9,40,6,37)(13,60,22,57,19,54,16,51)(14,53,23,50,20,59,17,56)(15,58,24,55,21,52,18,49)(25,61,34,70,31,67,28,64)(26,66,35,63,32,72,29,69)(27,71,36,68,33,65,30,62) );
G=PermutationGroup([[(1,33,23),(2,34,24),(3,35,13),(4,36,14),(5,25,15),(6,26,16),(7,27,17),(8,28,18),(9,29,19),(10,30,20),(11,31,21),(12,32,22),(37,66,51),(38,67,52),(39,68,53),(40,69,54),(41,70,55),(42,71,56),(43,72,57),(44,61,58),(45,62,59),(46,63,60),(47,64,49),(48,65,50)], [(1,25,19),(2,26,20),(3,27,21),(4,28,22),(5,29,23),(6,30,24),(7,31,13),(8,32,14),(9,33,15),(10,34,16),(11,35,17),(12,36,18),(37,62,55),(38,63,56),(39,64,57),(40,65,58),(41,66,59),(42,67,60),(43,68,49),(44,69,50),(45,70,51),(46,71,52),(47,72,53),(48,61,54)], [(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)], [(1,48,10,45,7,42,4,39),(2,41,11,38,8,47,5,44),(3,46,12,43,9,40,6,37),(13,60,22,57,19,54,16,51),(14,53,23,50,20,59,17,56),(15,58,24,55,21,52,18,49),(25,61,34,70,31,67,28,64),(26,66,35,63,32,72,29,69),(27,71,36,68,33,65,30,62)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6AQ | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12BH | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
162 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 | C32×C4.Dic3 | C32×C3⋊C8 | C3×C6×C12 | C3×C4.Dic3 | C32×C12 | C3×C62 | C3×C3⋊C8 | 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 | 8 | 2 | 2 | 16 | 8 | 16 | 16 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 8 | 8 | 4 | 16 | 32 |
Matrix representation of C32×C4.Dic3 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 8 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 46 | 0 |
G:=sub<GL(3,GF(73))| [64,0,0,0,1,0,0,0,1],[8,0,0,0,64,0,0,0,64],[1,0,0,0,46,0,0,0,27],[1,0,0,0,24,0,0,0,3],[1,0,0,0,0,46,0,1,0] >;
C32×C4.Dic3 in GAP, Magma, Sage, TeX
C_3^2\times C_4.{\rm Dic}_3 % in TeX
G:=Group("C3^2xC4.Dic3"); // GroupNames label
G:=SmallGroup(432,470);
// by ID
G=gap.SmallGroup(432,470);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,102,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=1,d^6=c^2,e^2=c^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations