direct product, metabelian, supersoluble, monomial
Aliases: C3xC12.53D4, C62.7Q8, C3:C8.1C12, C4.13(S3xC12), C12.5(C2xC12), C12.62(C3xD4), (C2xC6).6Dic6, C12.104(C4xS3), (C3xC12).164D4, (C2xC12).315D6, C4.Dic3.2C6, (C6xC12).43C22, C32:9(C8.C4), M4(2).1(C3xS3), (C3xM4(2)).9S3, (C3xM4(2)).7C6, C12.145(C3:D4), C22.1(C3xDic6), C6.24(Dic3:C4), (C32xM4(2)).1C2, (C2xC6).(C3xQ8), C6.8(C3xC4:C4), (C3xC3:C8).2C4, (C2xC3:C8).4C6, (C6xC3:C8).15C2, C3:2(C3xC8.C4), (C2xC4).36(S3xC6), C4.28(C3xC3:D4), (C3xC6).34(C4:C4), (C3xC12).41(C2xC4), (C2xC12).13(C2xC6), C2.5(C3xDic3:C4), (C3xC4.Dic3).6C2, SmallGroup(288,256)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC12.53D4
G = < a,b,c,d | a3=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c3 >
Subgroups: 122 in 70 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2xC4, C32, C12, C12, C2xC6, C2xC6, C2xC8, M4(2), M4(2), C3xC6, C3xC6, C3:C8, C3:C8, C24, C2xC12, C2xC12, C8.C4, C3xC12, C62, C2xC3:C8, C4.Dic3, C2xC24, C3xM4(2), C3xM4(2), C3xC3:C8, C3xC3:C8, C3xC24, C6xC12, C12.53D4, C3xC8.C4, C6xC3:C8, C3xC4.Dic3, C32xM4(2), C3xC12.53D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, Q8, C12, D6, C2xC6, C4:C4, C3xS3, Dic6, C4xS3, C3:D4, C2xC12, C3xD4, C3xQ8, C8.C4, S3xC6, Dic3:C4, C3xC4:C4, C3xDic6, S3xC12, C3xC3:D4, C12.53D4, C3xC8.C4, C3xDic3:C4, C3xC12.53D4
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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 33 4 36 7 27 10 30)(2 26 5 29 8 32 11 35)(3 31 6 34 9 25 12 28)(13 39 22 48 19 45 16 42)(14 44 23 41 20 38 17 47)(15 37 24 46 21 43 18 40)
(1 45 10 42 7 39 4 48)(2 38 11 47 8 44 5 41)(3 43 12 40 9 37 6 46)(13 36 22 33 19 30 16 27)(14 29 23 26 20 35 17 32)(15 34 24 31 21 28 18 25)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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), (1,33,4,36,7,27,10,30)(2,26,5,29,8,32,11,35)(3,31,6,34,9,25,12,28)(13,39,22,48,19,45,16,42)(14,44,23,41,20,38,17,47)(15,37,24,46,21,43,18,40), (1,45,10,42,7,39,4,48)(2,38,11,47,8,44,5,41)(3,43,12,40,9,37,6,46)(13,36,22,33,19,30,16,27)(14,29,23,26,20,35,17,32)(15,34,24,31,21,28,18,25)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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), (1,33,4,36,7,27,10,30)(2,26,5,29,8,32,11,35)(3,31,6,34,9,25,12,28)(13,39,22,48,19,45,16,42)(14,44,23,41,20,38,17,47)(15,37,24,46,21,43,18,40), (1,45,10,42,7,39,4,48)(2,38,11,47,8,44,5,41)(3,43,12,40,9,37,6,46)(13,36,22,33,19,30,16,27)(14,29,23,26,20,35,17,32)(15,34,24,31,21,28,18,25) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(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)], [(1,33,4,36,7,27,10,30),(2,26,5,29,8,32,11,35),(3,31,6,34,9,25,12,28),(13,39,22,48,19,45,16,42),(14,44,23,41,20,38,17,47),(15,37,24,46,21,43,18,40)], [(1,45,10,42,7,39,4,48),(2,38,11,47,8,44,5,41),(3,43,12,40,9,37,6,46),(13,36,22,33,19,30,16,27),(14,29,23,26,20,35,17,32),(15,34,24,31,21,28,18,25)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | 12N | 12O | 24A | ··· | 24P | 24Q | ··· | 24X | 24Y | 24Z | 24AA | 24AB |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
72 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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | Q8 | D6 | C3xS3 | C4xS3 | C3:D4 | C3xD4 | Dic6 | C3xQ8 | C8.C4 | S3xC6 | S3xC12 | C3xC3:D4 | C3xDic6 | C3xC8.C4 | C12.53D4 | C3xC12.53D4 |
| kernel | C3xC12.53D4 | C6xC3:C8 | C3xC4.Dic3 | C32xM4(2) | C12.53D4 | C3xC3:C8 | C2xC3:C8 | C4.Dic3 | C3xM4(2) | C3:C8 | C3xM4(2) | C3xC12 | C62 | C2xC12 | M4(2) | C12 | C12 | C12 | C2xC6 | C2xC6 | C32 | C2xC4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 2 | 4 |
Matrix representation of C3xC12.53D4 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
| 22 | 45 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 63 | 0 | 0 | 0 |
| 27 | 10 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[27,0,0,0,0,27,0,0,0,0,8,0,0,0,0,64],[22,0,0,0,45,10,0,0,0,0,0,1,0,0,1,0],[63,27,0,0,0,10,0,0,0,0,0,72,0,0,72,0] >;
C3xC12.53D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{53}D_4 % in TeX
G:=Group("C3xC12.53D4"); // GroupNames label
G:=SmallGroup(288,256);
// by ID
G=gap.SmallGroup(288,256);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,136,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^3>;
// generators/relations