Aliases: (C3xC24):2C4, C8:3(C32:C4), C32:2C8:5C4, (C3xC6).2C42, C32:4C8:10C4, C32:1(C8:C4), C3:S3.6M4(2), (C8xC3:S3).9C2, C2.3(C4xC32:C4), (C2xC32:C4).3C4, (C4xC32:C4).5C2, C4.18(C2xC32:C4), C3:S3:3C8.7C2, (C3xC12).11(C2xC4), (C4xC3:S3).79C22, C3:Dic3.26(C2xC4), (C2xC3:S3).23(C2xC4), SmallGroup(288,415)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C3xC24):C4
G = < a,b,c | a3=b24=c4=1, ab=ba, cac-1=a-1b8, cbc-1=a-1b13 >
Subgroups: 288 in 62 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2xC4, C32, Dic3, C12, D6, C42, C2xC8, C3:S3, C3xC6, C3:C8, C24, C4xS3, C8:C4, C3:Dic3, C3xC12, C32:C4, C2xC3:S3, S3xC8, C32:4C8, C3xC24, C32:2C8, C4xC3:S3, C2xC32:C4, C8xC3:S3, C3:S3:3C8, C4xC32:C4, (C3xC24):C4
Quotients: C1, C2, C4, C22, C2xC4, C42, M4(2), C8:C4, C32:C4, C2xC32:C4, C4xC32:C4, (C3xC24):C4
►On 48 points
(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(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 42)(2 31 18 47)(3 44 11 28)(4 33)(5 46 21 38)(6 35 14 43)(7 48)(8 37 24 29)(9 26 17 34)(10 39)(12 41 20 25)(13 30)(15 32 23 40)(16 45)(19 36)(22 27)
G:=sub<Sym(48)| (25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (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), (1,42)(2,31,18,47)(3,44,11,28)(4,33)(5,46,21,38)(6,35,14,43)(7,48)(8,37,24,29)(9,26,17,34)(10,39)(12,41,20,25)(13,30)(15,32,23,40)(16,45)(19,36)(22,27)>;
G:=Group( (25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (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), (1,42)(2,31,18,47)(3,44,11,28)(4,33)(5,46,21,38)(6,35,14,43)(7,48)(8,37,24,29)(9,26,17,34)(10,39)(12,41,20,25)(13,30)(15,32,23,40)(16,45)(19,36)(22,27) );
G=PermutationGroup([[(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(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)], [(1,42),(2,31,18,47),(3,44,11,28),(4,33),(5,46,21,38),(6,35,14,43),(7,48),(8,37,24,29),(9,26,17,34),(10,39),(12,41,20,25),(13,30),(15,32,23,40),(16,45),(19,36),(22,27)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 8A | 8B | 8C | ··· | 8H | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 1 | 1 | 9 | 9 | 18 | 18 | 18 | 18 | 4 | 4 | 2 | 2 | 18 | ··· | 18 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | M4(2) | C32:C4 | C2xC32:C4 | C4xC32:C4 | (C3xC24):C4 |
| kernel | (C3xC24):C4 | C8xC3:S3 | C3:S3:3C8 | C4xC32:C4 | C32:4C8 | C3xC24 | C32:2C8 | C2xC32:C4 | C3:S3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 8 |
Matrix representation of (C3xC24):C4 ►in GL6(F73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 51 | 8 | 0 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 18 | 24 | 0 | 0 | 0 | 0 |
| 26 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[51,0,0,0,0,0,8,22,0,0,0,0,0,0,27,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,27],[18,26,0,0,0,0,24,55,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
(C3xC24):C4 in GAP, Magma, Sage, TeX
(C_3\times C_{24})\rtimes C_4 % in TeX
G:=Group("(C3xC24):C4"); // GroupNames label
G:=SmallGroup(288,415);
// by ID
G=gap.SmallGroup(288,415);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c|a^3=b^24=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^8,c*b*c^-1=a^-1*b^13>;
// generators/relations