direct product, metacyclic, supersoluble, monomial
Aliases: C5×C4⋊Dic5, C20⋊3C20, C20⋊5Dic5, C10.22D20, C10.9Dic10, C102.18C22, C4⋊(C5×Dic5), (C5×C20)⋊10C4, C10.4(C5×D4), C2.1(C5×D20), (C5×C10).5Q8, C10.2(C5×Q8), C52⋊12(C4⋊C4), (C2×C20).5C10, (C10×C20).8C2, (C2×C20).19D5, (C5×C10).20D4, C10.15(C2×C20), (C2×C10).42D10, C2.2(C5×Dic10), C2.4(C10×Dic5), C22.5(D5×C10), (C2×Dic5).2C10, (C10×Dic5).2C2, C10.27(C2×Dic5), C5⋊3(C5×C4⋊C4), (C2×C4).3(C5×D5), (C5×C10).62(C2×C4), (C2×C10).7(C2×C10), SmallGroup(400,85)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C4⋊Dic5
G = < a,b,c,d | a5=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 148 in 68 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C4, C4, C22, C5, C5, C2×C4, C2×C4, C10, C10, C4⋊C4, Dic5, C20, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C5×C10, C4⋊Dic5, C5×C4⋊C4, C5×Dic5, C5×C20, C102, C10×Dic5, C10×C20, C5×C4⋊Dic5
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, D5, C10, C4⋊C4, Dic5, C20, D10, C2×C10, Dic10, D20, C2×Dic5, C2×C20, C5×D4, C5×Q8, C5×D5, C4⋊Dic5, C5×C4⋊C4, C5×Dic5, D5×C10, C5×Dic10, C5×D20, C10×Dic5, C5×C4⋊Dic5
►On 80 points
(1 7 3 9 5)(2 8 4 10 6)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 37 33 39 35)(32 38 34 40 36)(41 47 43 49 45)(42 48 44 50 46)(51 55 59 53 57)(52 56 60 54 58)(61 65 69 63 67)(62 66 70 64 68)(71 75 79 73 77)(72 76 80 74 78)
(1 50 27 40)(2 41 28 31)(3 42 29 32)(4 43 30 33)(5 44 21 34)(6 45 22 35)(7 46 23 36)(8 47 24 37)(9 48 25 38)(10 49 26 39)(11 60 80 68)(12 51 71 69)(13 52 72 70)(14 53 73 61)(15 54 74 62)(16 55 75 63)(17 56 76 64)(18 57 77 65)(19 58 78 66)(20 59 79 67)
(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)
(1 55 6 60)(2 54 7 59)(3 53 8 58)(4 52 9 57)(5 51 10 56)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
G:=sub<Sym(80)| (1,7,3,9,5)(2,8,4,10,6)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36)(41,47,43,49,45)(42,48,44,50,46)(51,55,59,53,57)(52,56,60,54,58)(61,65,69,63,67)(62,66,70,64,68)(71,75,79,73,77)(72,76,80,74,78), (1,50,27,40)(2,41,28,31)(3,42,29,32)(4,43,30,33)(5,44,21,34)(6,45,22,35)(7,46,23,36)(8,47,24,37)(9,48,25,38)(10,49,26,39)(11,60,80,68)(12,51,71,69)(13,52,72,70)(14,53,73,61)(15,54,74,62)(16,55,75,63)(17,56,76,64)(18,57,77,65)(19,58,78,66)(20,59,79,67), (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), (1,55,6,60)(2,54,7,59)(3,53,8,58)(4,52,9,57)(5,51,10,56)(11,50,16,45)(12,49,17,44)(13,48,18,43)(14,47,19,42)(15,46,20,41)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)>;
G:=Group( (1,7,3,9,5)(2,8,4,10,6)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,37,33,39,35)(32,38,34,40,36)(41,47,43,49,45)(42,48,44,50,46)(51,55,59,53,57)(52,56,60,54,58)(61,65,69,63,67)(62,66,70,64,68)(71,75,79,73,77)(72,76,80,74,78), (1,50,27,40)(2,41,28,31)(3,42,29,32)(4,43,30,33)(5,44,21,34)(6,45,22,35)(7,46,23,36)(8,47,24,37)(9,48,25,38)(10,49,26,39)(11,60,80,68)(12,51,71,69)(13,52,72,70)(14,53,73,61)(15,54,74,62)(16,55,75,63)(17,56,76,64)(18,57,77,65)(19,58,78,66)(20,59,79,67), (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), (1,55,6,60)(2,54,7,59)(3,53,8,58)(4,52,9,57)(5,51,10,56)(11,50,16,45)(12,49,17,44)(13,48,18,43)(14,47,19,42)(15,46,20,41)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75) );
G=PermutationGroup([[(1,7,3,9,5),(2,8,4,10,6),(11,15,19,13,17),(12,16,20,14,18),(21,27,23,29,25),(22,28,24,30,26),(31,37,33,39,35),(32,38,34,40,36),(41,47,43,49,45),(42,48,44,50,46),(51,55,59,53,57),(52,56,60,54,58),(61,65,69,63,67),(62,66,70,64,68),(71,75,79,73,77),(72,76,80,74,78)], [(1,50,27,40),(2,41,28,31),(3,42,29,32),(4,43,30,33),(5,44,21,34),(6,45,22,35),(7,46,23,36),(8,47,24,37),(9,48,25,38),(10,49,26,39),(11,60,80,68),(12,51,71,69),(13,52,72,70),(14,53,73,61),(15,54,74,62),(16,55,75,63),(17,56,76,64),(18,57,77,65),(19,58,78,66),(20,59,79,67)], [(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)], [(1,55,6,60),(2,54,7,59),(3,53,8,58),(4,52,9,57),(5,51,10,56),(11,50,16,45),(12,49,17,44),(13,48,18,43),(14,47,19,42),(15,46,20,41),(21,69,26,64),(22,68,27,63),(23,67,28,62),(24,66,29,61),(25,65,30,70),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10L | 10M | ··· | 10AP | 20A | ··· | 20AV | 20AW | ··· | 20BL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D4 | Q8 | D5 | Dic5 | D10 | Dic10 | D20 | C5×D4 | C5×Q8 | C5×D5 | C5×Dic5 | D5×C10 | C5×Dic10 | C5×D20 |
| kernel | C5×C4⋊Dic5 | C10×Dic5 | C10×C20 | C5×C20 | C4⋊Dic5 | C2×Dic5 | C2×C20 | C20 | C5×C10 | C5×C10 | C2×C20 | C20 | C2×C10 | C10 | C10 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 8 | 16 | 8 | 16 | 16 |
Matrix representation of C5×C4⋊Dic5 ►in GL3(𝔽41) generated by
| 16 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 37 |
| 40 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 32 |
| 40 | 0 | 0 |
| 0 | 37 | 0 |
| 0 | 0 | 10 |
| 9 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(41))| [16,0,0,0,37,0,0,0,37],[40,0,0,0,9,0,0,0,32],[40,0,0,0,37,0,0,0,10],[9,0,0,0,0,1,0,1,0] >;
C5×C4⋊Dic5 in GAP, Magma, Sage, TeX
C_5\times C_4\rtimes {\rm Dic}_5 % in TeX
G:=Group("C5xC4:Dic5"); // GroupNames label
G:=SmallGroup(400,85);
// by ID
G=gap.SmallGroup(400,85);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,247,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations