metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4.12D20, C20.47D4, 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 C4.12D20
G = < a,b,c | a20=1, b4=c2=a10, bab-1=cac-1=a-1, cbc-1=a5b3 >
Smallest permutation representation of C4.12D20
►On 80 points
Generators in S80
(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 71 6 66 11 61 16 76)(2 70 7 65 12 80 17 75)(3 69 8 64 13 79 18 74)(4 68 9 63 14 78 19 73)(5 67 10 62 15 77 20 72)(21 43 36 48 31 53 26 58)(22 42 37 47 32 52 27 57)(23 41 38 46 33 51 28 56)(24 60 39 45 34 50 29 55)(25 59 40 44 35 49 30 54)
(1 43 11 53)(2 42 12 52)(3 41 13 51)(4 60 14 50)(5 59 15 49)(6 58 16 48)(7 57 17 47)(8 56 18 46)(9 55 19 45)(10 54 20 44)(21 66 31 76)(22 65 32 75)(23 64 33 74)(24 63 34 73)(25 62 35 72)(26 61 36 71)(27 80 37 70)(28 79 38 69)(29 78 39 68)(30 77 40 67)
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), (1,71,6,66,11,61,16,76)(2,70,7,65,12,80,17,75)(3,69,8,64,13,79,18,74)(4,68,9,63,14,78,19,73)(5,67,10,62,15,77,20,72)(21,43,36,48,31,53,26,58)(22,42,37,47,32,52,27,57)(23,41,38,46,33,51,28,56)(24,60,39,45,34,50,29,55)(25,59,40,44,35,49,30,54), (1,43,11,53)(2,42,12,52)(3,41,13,51)(4,60,14,50)(5,59,15,49)(6,58,16,48)(7,57,17,47)(8,56,18,46)(9,55,19,45)(10,54,20,44)(21,66,31,76)(22,65,32,75)(23,64,33,74)(24,63,34,73)(25,62,35,72)(26,61,36,71)(27,80,37,70)(28,79,38,69)(29,78,39,68)(30,77,40,67)>;
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), (1,71,6,66,11,61,16,76)(2,70,7,65,12,80,17,75)(3,69,8,64,13,79,18,74)(4,68,9,63,14,78,19,73)(5,67,10,62,15,77,20,72)(21,43,36,48,31,53,26,58)(22,42,37,47,32,52,27,57)(23,41,38,46,33,51,28,56)(24,60,39,45,34,50,29,55)(25,59,40,44,35,49,30,54), (1,43,11,53)(2,42,12,52)(3,41,13,51)(4,60,14,50)(5,59,15,49)(6,58,16,48)(7,57,17,47)(8,56,18,46)(9,55,19,45)(10,54,20,44)(21,66,31,76)(22,65,32,75)(23,64,33,74)(24,63,34,73)(25,62,35,72)(26,61,36,71)(27,80,37,70)(28,79,38,69)(29,78,39,68)(30,77,40,67) );
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)], [(1,71,6,66,11,61,16,76),(2,70,7,65,12,80,17,75),(3,69,8,64,13,79,18,74),(4,68,9,63,14,78,19,73),(5,67,10,62,15,77,20,72),(21,43,36,48,31,53,26,58),(22,42,37,47,32,52,27,57),(23,41,38,46,33,51,28,56),(24,60,39,45,34,50,29,55),(25,59,40,44,35,49,30,54)], [(1,43,11,53),(2,42,12,52),(3,41,13,51),(4,60,14,50),(5,59,15,49),(6,58,16,48),(7,57,17,47),(8,56,18,46),(9,55,19,45),(10,54,20,44),(21,66,31,76),(22,65,32,75),(23,64,33,74),(24,63,34,73),(25,62,35,72),(26,61,36,71),(27,80,37,70),(28,79,38,69),(29,78,39,68),(30,77,40,67)]])
C4.12D20 is a maximal subgroup of
M4(2).19D10 D20.2D4 D5×C4.10D4 D20.7D4 D4.9D20 D4.10D20 C8.20D20 C8.24D20 M4(2).31D10 D4.3D20 D4.5D20 M4(2).13D10 D20.38D4 M4(2).16D10 D20.40D4 C12.6D20 C60.31D4 C4.D60
C4.12D20 is a maximal quotient of
C42.2D10 (C2×Dic5)⋊C8 C20.47D8 C20.2D8 M4(2)⋊Dic5 C12.6D20 C60.31D4 C4.D60
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 | C4.12D20 |
| kernel | C4.12D20 | 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 C4.12D20 ►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] >;
C4.12D20 in GAP, Magma, Sage, TeX
C_4._{12}D_{20} % in TeX
G:=Group("C4.12D20"); // 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
Export