direct product, metacyclic, nilpotent (class 2), monomial, 5-elementary
Aliases: C4×5- 1+2, C100⋊C5, C25⋊3C20, C52.C20, C50.2C10, C20.2C52, (C5×C20).C5, C5.2(C5×C20), C10.3(C5×C10), (C5×C10).3C10, C2.(C2×5- 1+2), (C2×5- 1+2).2C2, SmallGroup(500,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×5- 1+2
G = < a,b,c | a4=b25=c5=1, ab=ba, ac=ca, cbc-1=b6 >
Smallest permutation representation of C4×5- 1+2
►On 100 points
Generators in S100
(1 68 30 84)(2 69 31 85)(3 70 32 86)(4 71 33 87)(5 72 34 88)(6 73 35 89)(7 74 36 90)(8 75 37 91)(9 51 38 92)(10 52 39 93)(11 53 40 94)(12 54 41 95)(13 55 42 96)(14 56 43 97)(15 57 44 98)(16 58 45 99)(17 59 46 100)(18 60 47 76)(19 61 48 77)(20 62 49 78)(21 63 50 79)(22 64 26 80)(23 65 27 81)(24 66 28 82)(25 67 29 83)
(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 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 6 11 16 21)(3 23 18 13 8)(4 19 9 24 14)(5 15 25 10 20)(27 47 42 37 32)(28 43 33 48 38)(29 39 49 34 44)(30 35 40 45 50)(51 66 56 71 61)(52 62 72 57 67)(53 58 63 68 73)(55 75 70 65 60)(76 96 91 86 81)(77 92 82 97 87)(78 88 98 83 93)(79 84 89 94 99)
G:=sub<Sym(100)| (1,68,30,84)(2,69,31,85)(3,70,32,86)(4,71,33,87)(5,72,34,88)(6,73,35,89)(7,74,36,90)(8,75,37,91)(9,51,38,92)(10,52,39,93)(11,53,40,94)(12,54,41,95)(13,55,42,96)(14,56,43,97)(15,57,44,98)(16,58,45,99)(17,59,46,100)(18,60,47,76)(19,61,48,77)(20,62,49,78)(21,63,50,79)(22,64,26,80)(23,65,27,81)(24,66,28,82)(25,67,29,83), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50)(51,66,56,71,61)(52,62,72,57,67)(53,58,63,68,73)(55,75,70,65,60)(76,96,91,86,81)(77,92,82,97,87)(78,88,98,83,93)(79,84,89,94,99)>;
G:=Group( (1,68,30,84)(2,69,31,85)(3,70,32,86)(4,71,33,87)(5,72,34,88)(6,73,35,89)(7,74,36,90)(8,75,37,91)(9,51,38,92)(10,52,39,93)(11,53,40,94)(12,54,41,95)(13,55,42,96)(14,56,43,97)(15,57,44,98)(16,58,45,99)(17,59,46,100)(18,60,47,76)(19,61,48,77)(20,62,49,78)(21,63,50,79)(22,64,26,80)(23,65,27,81)(24,66,28,82)(25,67,29,83), (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(3,23,18,13,8)(4,19,9,24,14)(5,15,25,10,20)(27,47,42,37,32)(28,43,33,48,38)(29,39,49,34,44)(30,35,40,45,50)(51,66,56,71,61)(52,62,72,57,67)(53,58,63,68,73)(55,75,70,65,60)(76,96,91,86,81)(77,92,82,97,87)(78,88,98,83,93)(79,84,89,94,99) );
G=PermutationGroup([[(1,68,30,84),(2,69,31,85),(3,70,32,86),(4,71,33,87),(5,72,34,88),(6,73,35,89),(7,74,36,90),(8,75,37,91),(9,51,38,92),(10,52,39,93),(11,53,40,94),(12,54,41,95),(13,55,42,96),(14,56,43,97),(15,57,44,98),(16,58,45,99),(17,59,46,100),(18,60,47,76),(19,61,48,77),(20,62,49,78),(21,63,50,79),(22,64,26,80),(23,65,27,81),(24,66,28,82),(25,67,29,83)], [(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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,6,11,16,21),(3,23,18,13,8),(4,19,9,24,14),(5,15,25,10,20),(27,47,42,37,32),(28,43,33,48,38),(29,39,49,34,44),(30,35,40,45,50),(51,66,56,71,61),(52,62,72,57,67),(53,58,63,68,73),(55,75,70,65,60),(76,96,91,86,81),(77,92,82,97,87),(78,88,98,83,93),(79,84,89,94,99)]])
116 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20H | 20I | ··· | 20P | 25A | ··· | 25T | 50A | ··· | 50T | 100A | ··· | 100AN |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 5 | ··· | 5 | 5 | ··· | 5 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 |
| type | + | + | ||||||||||
| image | C1 | C2 | C4 | C5 | C5 | C10 | C10 | C20 | C20 | 5- 1+2 | C2×5- 1+2 | C4×5- 1+2 |
| kernel | C4×5- 1+2 | C2×5- 1+2 | 5- 1+2 | C100 | C5×C20 | C50 | C5×C10 | C25 | C52 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 20 | 4 | 20 | 4 | 40 | 8 | 4 | 4 | 8 |
Matrix representation of C4×5- 1+2 ►in GL5(𝔽101)
| 91 | 0 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 | 0 |
| 0 | 0 | 91 | 0 | 0 |
| 0 | 0 | 0 | 91 | 0 |
| 0 | 0 | 0 | 0 | 91 |
| 95 | 71 | 0 | 22 | 14 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 0 | 95 |
| 83 | 17 | 14 | 65 | 6 |
| 95 | 6 | 5 | 22 | 36 |
| 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 87 | 0 | 0 |
| 0 | 0 | 0 | 84 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(101))| [91,0,0,0,0,0,91,0,0,0,0,0,91,0,0,0,0,0,91,0,0,0,0,0,91],[95,0,0,0,83,71,0,0,0,17,0,1,0,0,14,22,0,87,0,65,14,0,0,95,6],[95,0,0,0,0,6,36,0,0,0,5,0,87,0,0,22,0,0,84,0,36,0,0,0,1] >;
C4×5- 1+2 in GAP, Magma, Sage, TeX
C_4\times 5_-^{1+2} % in TeX
G:=Group("C4xES-(5,1)"); // GroupNames label
G:=SmallGroup(500,14);
// by ID
G=gap.SmallGroup(500,14);
# by ID
G:=PCGroup([5,-2,-5,-5,-2,-5,250,506,3082]);
// Polycyclic
G:=Group<a,b,c|a^4=b^25=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations
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