metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.47D4, C4.12D20, M4(2).2D5, (C2×C4).2D10, C22.5(C4×D5), C5⋊2(C4.10D4), C4.22(C5⋊D4), (C2×Dic5).1C4, C4.Dic5.3C2, (C2×C20).14C22, (C2×Dic10).7C2, (C5×M4(2)).2C2, C10.20(C22⋊C4), C2.10(D10⋊C4), (C2×C10).23(C2×C4), SmallGroup(160,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.47D4
G = < a,b,c | a20=1, b4=c2=a10, bab-1=cac-1=a-1, cbc-1=a5b3 >
(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 62 6 77 11 72 16 67)(2 61 7 76 12 71 17 66)(3 80 8 75 13 70 18 65)(4 79 9 74 14 69 19 64)(5 78 10 73 15 68 20 63)(21 55 36 60 31 45 26 50)(22 54 37 59 32 44 27 49)(23 53 38 58 33 43 28 48)(24 52 39 57 34 42 29 47)(25 51 40 56 35 41 30 46)
(1 50 11 60)(2 49 12 59)(3 48 13 58)(4 47 14 57)(5 46 15 56)(6 45 16 55)(7 44 17 54)(8 43 18 53)(9 42 19 52)(10 41 20 51)(21 62 31 72)(22 61 32 71)(23 80 33 70)(24 79 34 69)(25 78 35 68)(26 77 36 67)(27 76 37 66)(28 75 38 65)(29 74 39 64)(30 73 40 63)
G:=sub<Sym(80)| (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), (1,62,6,77,11,72,16,67)(2,61,7,76,12,71,17,66)(3,80,8,75,13,70,18,65)(4,79,9,74,14,69,19,64)(5,78,10,73,15,68,20,63)(21,55,36,60,31,45,26,50)(22,54,37,59,32,44,27,49)(23,53,38,58,33,43,28,48)(24,52,39,57,34,42,29,47)(25,51,40,56,35,41,30,46), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,62,31,72)(22,61,32,71)(23,80,33,70)(24,79,34,69)(25,78,35,68)(26,77,36,67)(27,76,37,66)(28,75,38,65)(29,74,39,64)(30,73,40,63)>;
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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,62,6,77,11,72,16,67)(2,61,7,76,12,71,17,66)(3,80,8,75,13,70,18,65)(4,79,9,74,14,69,19,64)(5,78,10,73,15,68,20,63)(21,55,36,60,31,45,26,50)(22,54,37,59,32,44,27,49)(23,53,38,58,33,43,28,48)(24,52,39,57,34,42,29,47)(25,51,40,56,35,41,30,46), (1,50,11,60)(2,49,12,59)(3,48,13,58)(4,47,14,57)(5,46,15,56)(6,45,16,55)(7,44,17,54)(8,43,18,53)(9,42,19,52)(10,41,20,51)(21,62,31,72)(22,61,32,71)(23,80,33,70)(24,79,34,69)(25,78,35,68)(26,77,36,67)(27,76,37,66)(28,75,38,65)(29,74,39,64)(30,73,40,63) );
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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,62,6,77,11,72,16,67),(2,61,7,76,12,71,17,66),(3,80,8,75,13,70,18,65),(4,79,9,74,14,69,19,64),(5,78,10,73,15,68,20,63),(21,55,36,60,31,45,26,50),(22,54,37,59,32,44,27,49),(23,53,38,58,33,43,28,48),(24,52,39,57,34,42,29,47),(25,51,40,56,35,41,30,46)], [(1,50,11,60),(2,49,12,59),(3,48,13,58),(4,47,14,57),(5,46,15,56),(6,45,16,55),(7,44,17,54),(8,43,18,53),(9,42,19,52),(10,41,20,51),(21,62,31,72),(22,61,32,71),(23,80,33,70),(24,79,34,69),(25,78,35,68),(26,77,36,67),(27,76,37,66),(28,75,38,65),(29,74,39,64),(30,73,40,63)])
31 conjugacy classes
class | 1 | 2A | 2B | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
order | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
31 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
type | + | + | + | + | + | + | + | + | - | - | |||
image | C1 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | D20 | C5⋊D4 | C4×D5 | C4.10D4 | C20.47D4 |
kernel | C20.47D4 | C4.Dic5 | C5×M4(2) | C2×Dic10 | C2×Dic5 | C20 | M4(2) | C2×C4 | C4 | C4 | C22 | C5 | C1 |
# reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 4 |
Matrix representation of C20.47D4 ►in GL4(𝔽41) generated by
25 | 39 | 0 | 0 |
2 | 13 | 0 | 0 |
0 | 0 | 25 | 39 |
0 | 0 | 2 | 13 |
0 | 0 | 14 | 37 |
0 | 0 | 39 | 27 |
2 | 15 | 0 | 0 |
27 | 39 | 0 | 0 |
14 | 37 | 0 | 0 |
39 | 27 | 0 | 0 |
0 | 0 | 14 | 37 |
0 | 0 | 39 | 27 |
G:=sub<GL(4,GF(41))| [25,2,0,0,39,13,0,0,0,0,25,2,0,0,39,13],[0,0,2,27,0,0,15,39,14,39,0,0,37,27,0,0],[14,39,0,0,37,27,0,0,0,0,14,39,0,0,37,27] >;
C20.47D4 in GAP, Magma, Sage, TeX
C_{20}._{47}D_4
% in TeX
G:=Group("C20.47D4");
// GroupNames label
G:=SmallGroup(160,31);
// by ID
G=gap.SmallGroup(160,31);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,121,31,362,86,297,4613]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^4=c^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^5*b^3>;
// generators/relations