direct product, abelian, monomial, 2-elementary
Aliases: C4×C20, SmallGroup(80,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C20 |
| C1 — C4×C20 |
| C1 — C4×C20 |
Generators and relations for C4×C20
G = < a,b | a4=b20=1, ab=ba >
Smallest permutation representation of C4×C20
►Regular action on 80 points
Generators in S80
(1 80 46 32)(2 61 47 33)(3 62 48 34)(4 63 49 35)(5 64 50 36)(6 65 51 37)(7 66 52 38)(8 67 53 39)(9 68 54 40)(10 69 55 21)(11 70 56 22)(12 71 57 23)(13 72 58 24)(14 73 59 25)(15 74 60 26)(16 75 41 27)(17 76 42 28)(18 77 43 29)(19 78 44 30)(20 79 45 31)
(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)
G:=sub<Sym(80)| (1,80,46,32)(2,61,47,33)(3,62,48,34)(4,63,49,35)(5,64,50,36)(6,65,51,37)(7,66,52,38)(8,67,53,39)(9,68,54,40)(10,69,55,21)(11,70,56,22)(12,71,57,23)(13,72,58,24)(14,73,59,25)(15,74,60,26)(16,75,41,27)(17,76,42,28)(18,77,43,29)(19,78,44,30)(20,79,45,31), (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)>;
G:=Group( (1,80,46,32)(2,61,47,33)(3,62,48,34)(4,63,49,35)(5,64,50,36)(6,65,51,37)(7,66,52,38)(8,67,53,39)(9,68,54,40)(10,69,55,21)(11,70,56,22)(12,71,57,23)(13,72,58,24)(14,73,59,25)(15,74,60,26)(16,75,41,27)(17,76,42,28)(18,77,43,29)(19,78,44,30)(20,79,45,31), (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) );
G=PermutationGroup([[(1,80,46,32),(2,61,47,33),(3,62,48,34),(4,63,49,35),(5,64,50,36),(6,65,51,37),(7,66,52,38),(8,67,53,39),(9,68,54,40),(10,69,55,21),(11,70,56,22),(12,71,57,23),(13,72,58,24),(14,73,59,25),(15,74,60,26),(16,75,41,27),(17,76,42,28),(18,77,43,29),(19,78,44,30),(20,79,45,31)], [(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)]])
C4×C20 is a maximal subgroup of
C42.D5 C20⋊3C8 D20⋊4C4 C20⋊2Q8 C20.6Q8 C42⋊D5 C20⋊4D4 C4.D20 C42⋊2D5
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C5 | C10 | C20 |
| kernel | C4×C20 | C2×C20 | C20 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 4 | 12 | 48 |
Matrix representation of C4×C20 ►in GL2(𝔽41) generated by
| 40 | 0 |
| 0 | 32 |
| 8 | 0 |
| 0 | 16 |
G:=sub<GL(2,GF(41))| [40,0,0,32],[8,0,0,16] >;
C4×C20 in GAP, Magma, Sage, TeX
C_4\times C_{20} % in TeX
G:=Group("C4xC20"); // GroupNames label
G:=SmallGroup(80,20);
// by ID
G=gap.SmallGroup(80,20);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-2,100,206]);
// Polycyclic
G:=Group<a,b|a^4=b^20=1,a*b=b*a>;
// generators/relations
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