metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.42D4, C8.24D20, D20.22D4, Dic10.22D4, M4(2).11D10, (C2×C8).72D10, C4.58(C2×D20), C8.C4⋊7D5, C4.137(D4×D5), C20.138(C2×D4), C8⋊D10.2C2, C5⋊3(D4.3D4), D20.3C4⋊9C2, C20.47D4⋊4C2, C8.D10⋊10C2, C20.46D4⋊4C2, C2.24(C4⋊2D20), C10.51(C4⋊D4), (C2×C40).155C22, (C2×C20).314C23, 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 C40.42D4
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 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C5, C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×3], C23, D5 [×2], C10, C10, C2×C8, C2×C8, M4(2) [×2], M4(2) [×2], D8, SD16 [×4], Q16, C2×D4, C2×Q8, C4○D4, Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C5⋊2C8, C40 [×2], C40 [×2], Dic10, Dic10 [×2], C4×D5, D20, D20 [×2], C2×Dic5, C5⋊D4, C2×C20, C22×D5, D4.3D4, C8×D5, C8⋊D5, C40⋊C2 [×4], D40, Dic20, C4.Dic5, C2×C40, C5×M4(2) [×2], C2×Dic10, C2×D20, C4○D20, C20.46D4, C20.47D4, C5×C8.C4, D20.3C4, C2×C40⋊C2, C8⋊D10, C8.D10, C40.42D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, D20 [×2], C22×D5, D4.3D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊2D20, C40.42D4
►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 52 31 62 21 72 11 42)(2 63 32 73 22 43 12 53)(3 74 33 44 23 54 13 64)(4 45 34 55 24 65 14 75)(5 56 35 66 25 76 15 46)(6 67 36 77 26 47 16 57)(7 78 37 48 27 58 17 68)(8 49 38 59 28 69 18 79)(9 60 39 70 29 80 19 50)(10 71 40 41 30 51 20 61)
(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 78 61 58)(42 57 62 77)(43 76 63 56)(44 55 64 75)(45 74 65 54)(46 53 66 73)(47 72 67 52)(48 51 68 71)(49 70 69 50)(59 60 79 80)
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), (1,52,31,62,21,72,11,42)(2,63,32,73,22,43,12,53)(3,74,33,44,23,54,13,64)(4,45,34,55,24,65,14,75)(5,56,35,66,25,76,15,46)(6,67,36,77,26,47,16,57)(7,78,37,48,27,58,17,68)(8,49,38,59,28,69,18,79)(9,60,39,70,29,80,19,50)(10,71,40,41,30,51,20,61), (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)(41,78,61,58)(42,57,62,77)(43,76,63,56)(44,55,64,75)(45,74,65,54)(46,53,66,73)(47,72,67,52)(48,51,68,71)(49,70,69,50)(59,60,79,80)>;
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), (1,52,31,62,21,72,11,42)(2,63,32,73,22,43,12,53)(3,74,33,44,23,54,13,64)(4,45,34,55,24,65,14,75)(5,56,35,66,25,76,15,46)(6,67,36,77,26,47,16,57)(7,78,37,48,27,58,17,68)(8,49,38,59,28,69,18,79)(9,60,39,70,29,80,19,50)(10,71,40,41,30,51,20,61), (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)(41,78,61,58)(42,57,62,77)(43,76,63,56)(44,55,64,75)(45,74,65,54)(46,53,66,73)(47,72,67,52)(48,51,68,71)(49,70,69,50)(59,60,79,80) );
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)], [(1,52,31,62,21,72,11,42),(2,63,32,73,22,43,12,53),(3,74,33,44,23,54,13,64),(4,45,34,55,24,65,14,75),(5,56,35,66,25,76,15,46),(6,67,36,77,26,47,16,57),(7,78,37,48,27,58,17,68),(8,49,38,59,28,69,18,79),(9,60,39,70,29,80,19,50),(10,71,40,41,30,51,20,61)], [(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),(41,78,61,58),(42,57,62,77),(43,76,63,56),(44,55,64,75),(45,74,65,54),(46,53,66,73),(47,72,67,52),(48,51,68,71),(49,70,69,50),(59,60,79,80)])
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 | C40.42D4 |
| kernel | C40.42D4 | C20.46D4 | C20.47D4 | 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 C40.42D4 ►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] >;
C40.42D4 in GAP, Magma, Sage, TeX
C_{40}._{42}D_4 % in TeX
G:=Group("C40.42D4"); // 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