metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.24D20, C40.42D4, D20.22D4, Dic10.22D4, M4(2).11D10, C4.137(D4×D5), C8.C4⋊7D5, C4.58(C2×D20), (C2×C8).72D10, C8⋊D10.2C2, C20.138(C2×D4), C5⋊3(D4.3D4), D20.3C4⋊9C2, C4.12D20⋊4C2, C8.D10⋊10C2, C20.46D4⋊4C2, C10.51(C4⋊D4), C2.24(C4⋊D20), (C2×C20).314C23, (C2×C40).155C22, C4○D20.41C22, (C2×D20).93C22, C22.8(Q8⋊2D5), (C5×M4(2)).8C22, C4.Dic5.39C22, (C2×Dic10).99C22, (C5×C8.C4)⋊8C2, (C2×C40⋊C2)⋊26C2, (C2×C10).5(C4○D4), (C2×C4).115(C22×D5), SmallGroup(320,525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.24D20
G = < a,b,c | a40=1, b4=c2=a20, bab-1=a11, cac-1=a19, cbc-1=b3 >
Subgroups: 510 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C5⋊2C8, C40, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2, D40, Dic20, C4.Dic5, C2×C40, C5×M4(2), C2×Dic10, C2×D20, C4○D20, C20.46D4, C4.12D20, C5×C8.C4, D20.3C4, C2×C40⋊C2, C8⋊D10, C8.D10, C8.24D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C22×D5, D4.3D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, C8.24D20
►On 80 points
(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 67 31 77 21 47 11 57)(2 78 32 48 22 58 12 68)(3 49 33 59 23 69 13 79)(4 60 34 70 24 80 14 50)(5 71 35 41 25 51 15 61)(6 42 36 52 26 62 16 72)(7 53 37 63 27 73 17 43)(8 64 38 74 28 44 18 54)(9 75 39 45 29 55 19 65)(10 46 40 56 30 66 20 76)
(1 36 21 16)(2 15 22 35)(3 34 23 14)(4 13 24 33)(5 32 25 12)(6 11 26 31)(7 30 27 10)(8 9 28 29)(17 20 37 40)(18 39 38 19)(41 48 61 68)(42 67 62 47)(43 46 63 66)(44 65 64 45)(49 80 69 60)(50 59 70 79)(51 78 71 58)(52 57 72 77)(53 76 73 56)(54 55 74 75)
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), (1,67,31,77,21,47,11,57)(2,78,32,48,22,58,12,68)(3,49,33,59,23,69,13,79)(4,60,34,70,24,80,14,50)(5,71,35,41,25,51,15,61)(6,42,36,52,26,62,16,72)(7,53,37,63,27,73,17,43)(8,64,38,74,28,44,18,54)(9,75,39,45,29,55,19,65)(10,46,40,56,30,66,20,76), (1,36,21,16)(2,15,22,35)(3,34,23,14)(4,13,24,33)(5,32,25,12)(6,11,26,31)(7,30,27,10)(8,9,28,29)(17,20,37,40)(18,39,38,19)(41,48,61,68)(42,67,62,47)(43,46,63,66)(44,65,64,45)(49,80,69,60)(50,59,70,79)(51,78,71,58)(52,57,72,77)(53,76,73,56)(54,55,74,75)>;
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), (1,67,31,77,21,47,11,57)(2,78,32,48,22,58,12,68)(3,49,33,59,23,69,13,79)(4,60,34,70,24,80,14,50)(5,71,35,41,25,51,15,61)(6,42,36,52,26,62,16,72)(7,53,37,63,27,73,17,43)(8,64,38,74,28,44,18,54)(9,75,39,45,29,55,19,65)(10,46,40,56,30,66,20,76), (1,36,21,16)(2,15,22,35)(3,34,23,14)(4,13,24,33)(5,32,25,12)(6,11,26,31)(7,30,27,10)(8,9,28,29)(17,20,37,40)(18,39,38,19)(41,48,61,68)(42,67,62,47)(43,46,63,66)(44,65,64,45)(49,80,69,60)(50,59,70,79)(51,78,71,58)(52,57,72,77)(53,76,73,56)(54,55,74,75) );
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)], [(1,67,31,77,21,47,11,57),(2,78,32,48,22,58,12,68),(3,49,33,59,23,69,13,79),(4,60,34,70,24,80,14,50),(5,71,35,41,25,51,15,61),(6,42,36,52,26,62,16,72),(7,53,37,63,27,73,17,43),(8,64,38,74,28,44,18,54),(9,75,39,45,29,55,19,65),(10,46,40,56,30,66,20,76)], [(1,36,21,16),(2,15,22,35),(3,34,23,14),(4,13,24,33),(5,32,25,12),(6,11,26,31),(7,30,27,10),(8,9,28,29),(17,20,37,40),(18,39,38,19),(41,48,61,68),(42,67,62,47),(43,46,63,66),(44,65,64,45),(49,80,69,60),(50,59,70,79),(51,78,71,58),(52,57,72,77),(53,76,73,56),(54,55,74,75)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 40 | 2 | 2 | 20 | 40 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | D4.3D4 | D4×D5 | Q8⋊2D5 | C8.24D20 |
| kernel | C8.24D20 | C20.46D4 | C4.12D20 | C5×C8.C4 | D20.3C4 | C2×C40⋊C2 | C8⋊D10 | C8.D10 | C40 | Dic10 | D20 | C8.C4 | C2×C10 | C2×C8 | M4(2) | C8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C8.24D20 ►in GL4(𝔽41) generated by
| 39 | 14 | 0 | 0 |
| 27 | 37 | 0 | 0 |
| 0 | 0 | 27 | 2 |
| 0 | 0 | 39 | 15 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 28 | 0 | 0 |
| 13 | 39 | 0 | 0 |
| 15 | 39 | 0 | 0 |
| 31 | 26 | 0 | 0 |
| 0 | 0 | 4 | 14 |
| 0 | 0 | 31 | 37 |
G:=sub<GL(4,GF(41))| [39,27,0,0,14,37,0,0,0,0,27,39,0,0,2,15],[0,0,2,13,0,0,28,39,1,0,0,0,0,1,0,0],[15,31,0,0,39,26,0,0,0,0,4,31,0,0,14,37] >;
C8.24D20 in GAP, Magma, Sage, TeX
C_8._{24}D_{20} % in TeX
G:=Group("C8.24D20"); // GroupNames label
G:=SmallGroup(320,525);
// by ID
G=gap.SmallGroup(320,525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,1123,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=1,b^4=c^2=a^20,b*a*b^-1=a^11,c*a*c^-1=a^19,c*b*c^-1=b^3>;
// generators/relations