direct product, metabelian, soluble, monomial, A-group
Aliases: C10xC32:C4, (C3xC6):C20, C3:S3:2C20, (C3xC30):5C4, C32:1(C2xC20), (C5xC3:S3):7C4, (C3xC15):11(C2xC4), (C2xC3:S3).2C10, (C10xC3:S3).4C2, C3:S3.3(C2xC10), (C5xC3:S3).7C22, SmallGroup(360,148)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C10xC32:C4 |
Generators and relations for C10xC32:C4
G = < a,b,c,d | a10=b3=c3=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >
Quotients: C1, C2, C4, C22, C5, C2xC4, C10, C20, C2xC10, C32:C4, C2xC20, C2xC32:C4, C5xC32:C4, C10xC32:C4
9C2
9C2
2C3
2C3
9C4
9C4
9C22
2C6
2C6
6S3
6S3
6S3
6S3
9C10
9C10
2C15
2C15
6D6
6D6
9C20
9C20
2C30
2C30
Smallest permutation representation of C10xC32:C4
►On 60 points
Generators in S60
(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)
(11 57 26)(12 58 27)(13 59 28)(14 60 29)(15 51 30)(16 52 21)(17 53 22)(18 54 23)(19 55 24)(20 56 25)
(1 49 39)(2 50 40)(3 41 31)(4 42 32)(5 43 33)(6 44 34)(7 45 35)(8 46 36)(9 47 37)(10 48 38)(11 57 26)(12 58 27)(13 59 28)(14 60 29)(15 51 30)(16 52 21)(17 53 22)(18 54 23)(19 55 24)(20 56 25)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 45 26 35)(12 46 27 36)(13 47 28 37)(14 48 29 38)(15 49 30 39)(16 50 21 40)(17 41 22 31)(18 42 23 32)(19 43 24 33)(20 44 25 34)
G:=sub<Sym(60)| (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), (11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,51,30)(16,52,21)(17,53,22)(18,54,23)(19,55,24)(20,56,25), (1,49,39)(2,50,40)(3,41,31)(4,42,32)(5,43,33)(6,44,34)(7,45,35)(8,46,36)(9,47,37)(10,48,38)(11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,51,30)(16,52,21)(17,53,22)(18,54,23)(19,55,24)(20,56,25), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,45,26,35)(12,46,27,36)(13,47,28,37)(14,48,29,38)(15,49,30,39)(16,50,21,40)(17,41,22,31)(18,42,23,32)(19,43,24,33)(20,44,25,34)>;
G:=Group( (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), (11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,51,30)(16,52,21)(17,53,22)(18,54,23)(19,55,24)(20,56,25), (1,49,39)(2,50,40)(3,41,31)(4,42,32)(5,43,33)(6,44,34)(7,45,35)(8,46,36)(9,47,37)(10,48,38)(11,57,26)(12,58,27)(13,59,28)(14,60,29)(15,51,30)(16,52,21)(17,53,22)(18,54,23)(19,55,24)(20,56,25), (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,45,26,35)(12,46,27,36)(13,47,28,37)(14,48,29,38)(15,49,30,39)(16,50,21,40)(17,41,22,31)(18,42,23,32)(19,43,24,33)(20,44,25,34) );
G=PermutationGroup([[(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)], [(11,57,26),(12,58,27),(13,59,28),(14,60,29),(15,51,30),(16,52,21),(17,53,22),(18,54,23),(19,55,24),(20,56,25)], [(1,49,39),(2,50,40),(3,41,31),(4,42,32),(5,43,33),(6,44,34),(7,45,35),(8,46,36),(9,47,37),(10,48,38),(11,57,26),(12,58,27),(13,59,28),(14,60,29),(15,51,30),(16,52,21),(17,53,22),(18,54,23),(19,55,24),(20,56,25)], [(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,45,26,35),(12,46,27,36),(13,47,28,37),(14,48,29,38),(15,49,30,39),(16,50,21,40),(17,41,22,31),(18,42,23,32),(19,43,24,33),(20,44,25,34)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 15A | ··· | 15H | 20A | ··· | 20P | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 9 | 9 | 4 | 4 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 4 | ··· | 4 | 9 | ··· | 9 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | C32:C4 | C2xC32:C4 | C5xC32:C4 | C10xC32:C4 |
| kernel | C10xC32:C4 | C5xC32:C4 | C10xC3:S3 | C5xC3:S3 | C3xC30 | C2xC32:C4 | C32:C4 | C2xC3:S3 | C3:S3 | C3xC6 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 2 | 8 | 8 |
Matrix representation of C10xC32:C4 ►in GL4(F61) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 60 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
G:=sub<GL(4,GF(61))| [52,0,0,0,0,52,0,0,0,0,52,0,0,0,0,52],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,60],[60,1,0,0,60,0,0,0,0,0,0,60,0,0,1,60],[0,0,60,1,0,0,0,1,60,0,0,0,0,60,0,0] >;
C10xC32:C4 in GAP, Magma, Sage, TeX
C_{10}\times C_3^2\rtimes C_4 % in TeX
G:=Group("C10xC3^2:C4"); // GroupNames label
G:=SmallGroup(360,148);
// by ID
G=gap.SmallGroup(360,148);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-3,3,120,8404,142,11525,455]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations
Export