metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42:3Dic3, C12.19C42, (C4xC12):1C4, (C2xC12).6Q8, (C4xDic3):3C4, (C2xC4).13D12, C12.28(C4:C4), (C2xC4).2Dic6, C4.33(D6:C4), C3:1(C4.9C42), (C2xC12).104D4, C4.24(C4xDic3), (C22xC6).42D4, (C22xC4).73D6, C12.9(C22:C4), C4.8(Dic3:C4), C42:C2.1S3, C4.12(C4:Dic3), C22.17(D6:C4), C23.23(C3:D4), C6.7(C2.C42), C2.8(C6.C42), C22.3(Dic3:C4), C23.26D6.7C2, (C22xC12).120C22, C22.10(C6.D4), (C2xC3:C8):1C4, (C2xC6).3(C4:C4), (C2xC12).56(C2xC4), (C2xC4).139(C4xS3), (C2xC4).20(C3:D4), (C2xC4).72(C2xDic3), (C2xC4.Dic3).7C2, (C2xC6).89(C22:C4), (C3xC42:C2).1C2, SmallGroup(192,90)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42:3Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 216 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, C23, Dic3, C12, C12, C2xC6, C2xC6, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C3:C8, C2xDic3, C2xC12, C2xC12, C22xC6, C42:C2, C42:C2, C2xM4(2), C2xC3:C8, C4.Dic3, C4xDic3, C4:Dic3, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C4.9C42, C2xC4.Dic3, C23.26D6, C3xC42:C2, C42:3Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Q8, Dic3, D6, C42, C22:C4, C4:C4, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2.C42, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C4.9C42, C6.C42, C42:3Dic3
►On 48 points
(1 36 22 42)(2 34 23 40)(3 32 24 38)(4 26 18 47)(5 30 16 45)(6 28 17 43)(7 29 14 44)(8 27 15 48)(9 25 13 46)(10 33 21 39)(11 31 19 37)(12 35 20 41)
(1 6 10 9)(2 4 11 7)(3 5 12 8)(13 22 17 21)(14 23 18 19)(15 24 16 20)(25 36 28 33)(26 31 29 34)(27 32 30 35)(37 44 40 47)(38 45 41 48)(39 46 42 43)
(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)
(1 17)(2 16)(3 18)(4 20)(5 19)(6 21)(7 24)(8 23)(9 22)(10 13)(11 15)(12 14)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 38 34 41)(32 37 35 40)(33 42 36 39)
G:=sub<Sym(48)| (1,36,22,42)(2,34,23,40)(3,32,24,38)(4,26,18,47)(5,30,16,45)(6,28,17,43)(7,29,14,44)(8,27,15,48)(9,25,13,46)(10,33,21,39)(11,31,19,37)(12,35,20,41), (1,6,10,9)(2,4,11,7)(3,5,12,8)(13,22,17,21)(14,23,18,19)(15,24,16,20)(25,36,28,33)(26,31,29,34)(27,32,30,35)(37,44,40,47)(38,45,41,48)(39,46,42,43), (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), (1,17)(2,16)(3,18)(4,20)(5,19)(6,21)(7,24)(8,23)(9,22)(10,13)(11,15)(12,14)(25,43,28,46)(26,48,29,45)(27,47,30,44)(31,38,34,41)(32,37,35,40)(33,42,36,39)>;
G:=Group( (1,36,22,42)(2,34,23,40)(3,32,24,38)(4,26,18,47)(5,30,16,45)(6,28,17,43)(7,29,14,44)(8,27,15,48)(9,25,13,46)(10,33,21,39)(11,31,19,37)(12,35,20,41), (1,6,10,9)(2,4,11,7)(3,5,12,8)(13,22,17,21)(14,23,18,19)(15,24,16,20)(25,36,28,33)(26,31,29,34)(27,32,30,35)(37,44,40,47)(38,45,41,48)(39,46,42,43), (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), (1,17)(2,16)(3,18)(4,20)(5,19)(6,21)(7,24)(8,23)(9,22)(10,13)(11,15)(12,14)(25,43,28,46)(26,48,29,45)(27,47,30,44)(31,38,34,41)(32,37,35,40)(33,42,36,39) );
G=PermutationGroup([[(1,36,22,42),(2,34,23,40),(3,32,24,38),(4,26,18,47),(5,30,16,45),(6,28,17,43),(7,29,14,44),(8,27,15,48),(9,25,13,46),(10,33,21,39),(11,31,19,37),(12,35,20,41)], [(1,6,10,9),(2,4,11,7),(3,5,12,8),(13,22,17,21),(14,23,18,19),(15,24,16,20),(25,36,28,33),(26,31,29,34),(27,32,30,35),(37,44,40,47),(38,45,41,48),(39,46,42,43)], [(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)], [(1,17),(2,16),(3,18),(4,20),(5,19),(6,21),(7,24),(8,23),(9,22),(10,13),(11,15),(12,14),(25,43,28,46),(26,48,29,45),(27,47,30,44),(31,38,34,41),(32,37,35,40),(33,42,36,39)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | D4 | Dic3 | D6 | Dic6 | C4xS3 | D12 | C3:D4 | C3:D4 | C4.9C42 | C42:3Dic3 |
| kernel | C42:3Dic3 | C2xC4.Dic3 | C23.26D6 | C3xC42:C2 | C2xC3:C8 | C4xDic3 | C4xC12 | C42:C2 | C2xC12 | C2xC12 | C22xC6 | C42 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 |
Matrix representation of C42:3Dic3 ►in GL4(F73) generated by
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 72 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 13 | 43 | 0 | 0 |
| 30 | 60 | 0 | 0 |
| 0 | 0 | 59 | 66 |
| 0 | 0 | 7 | 14 |
G:=sub<GL(4,GF(73))| [0,0,30,13,0,0,60,43,1,0,0,0,0,1,0,0],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,1,0,0,72,1,0,0,0,0,0,72,0,0,1,72],[13,30,0,0,43,60,0,0,0,0,59,7,0,0,66,14] >;
C42:3Dic3 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3{\rm Dic}_3 % in TeX
G:=Group("C4^2:3Dic3"); // GroupNames label
G:=SmallGroup(192,90);
// by ID
G=gap.SmallGroup(192,90);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations