metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1D10, C20.13D4, C4.15D20, Dic20⋊2C2, M4(2)⋊2D5, C40.1C22, C22.6D20, C20.33C23, D20.8C22, Dic10.8C22, C40⋊C2⋊2C2, (C2×C10).6D4, C4○D20.4C2, (C2×C4).16D10, C2.16(C2×D20), C10.14(C2×D4), C5⋊1(C8.C22), (C2×Dic10)⋊8C2, (C5×M4(2))⋊2C2, C4.31(C22×D5), (C2×C20).28C22, SmallGroup(160,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D10
G = < a,b,c | a8=1, b10=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b9 >
Subgroups: 208 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C8.C22, C40, Dic10, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, C8.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8.C22, D20, C22×D5, C2×D20, C8.D10
►On 80 points
(1 68 23 45 11 78 33 55)(2 79 24 56 12 69 34 46)(3 70 25 47 13 80 35 57)(4 61 26 58 14 71 36 48)(5 72 27 49 15 62 37 59)(6 63 28 60 16 73 38 50)(7 74 29 51 17 64 39 41)(8 65 30 42 18 75 40 52)(9 76 31 53 19 66 21 43)(10 67 32 44 20 77 22 54)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 24 31 34)(22 33 32 23)(25 40 35 30)(26 29 36 39)(27 38 37 28)(41 61 51 71)(42 70 52 80)(43 79 53 69)(44 68 54 78)(45 77 55 67)(46 66 56 76)(47 75 57 65)(48 64 58 74)(49 73 59 63)(50 62 60 72)
G:=sub<Sym(80)| (1,68,23,45,11,78,33,55)(2,79,24,56,12,69,34,46)(3,70,25,47,13,80,35,57)(4,61,26,58,14,71,36,48)(5,72,27,49,15,62,37,59)(6,63,28,60,16,73,38,50)(7,74,29,51,17,64,39,41)(8,65,30,42,18,75,40,52)(9,76,31,53,19,66,21,43)(10,67,32,44,20,77,22,54), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,61,51,71)(42,70,52,80)(43,79,53,69)(44,68,54,78)(45,77,55,67)(46,66,56,76)(47,75,57,65)(48,64,58,74)(49,73,59,63)(50,62,60,72)>;
G:=Group( (1,68,23,45,11,78,33,55)(2,79,24,56,12,69,34,46)(3,70,25,47,13,80,35,57)(4,61,26,58,14,71,36,48)(5,72,27,49,15,62,37,59)(6,63,28,60,16,73,38,50)(7,74,29,51,17,64,39,41)(8,65,30,42,18,75,40,52)(9,76,31,53,19,66,21,43)(10,67,32,44,20,77,22,54), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,24,31,34)(22,33,32,23)(25,40,35,30)(26,29,36,39)(27,38,37,28)(41,61,51,71)(42,70,52,80)(43,79,53,69)(44,68,54,78)(45,77,55,67)(46,66,56,76)(47,75,57,65)(48,64,58,74)(49,73,59,63)(50,62,60,72) );
G=PermutationGroup([[(1,68,23,45,11,78,33,55),(2,79,24,56,12,69,34,46),(3,70,25,47,13,80,35,57),(4,61,26,58,14,71,36,48),(5,72,27,49,15,62,37,59),(6,63,28,60,16,73,38,50),(7,74,29,51,17,64,39,41),(8,65,30,42,18,75,40,52),(9,76,31,53,19,66,21,43),(10,67,32,44,20,77,22,54)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,24,31,34),(22,33,32,23),(25,40,35,30),(26,29,36,39),(27,38,37,28),(41,61,51,71),(42,70,52,80),(43,79,53,69),(44,68,54,78),(45,77,55,67),(46,66,56,76),(47,75,57,65),(48,64,58,74),(49,73,59,63),(50,62,60,72)]])
C8.D10 is a maximal subgroup of
D20.1D4 D20.2D4 D20.4D4 D20.7D4 M4(2)⋊D10 D4.9D20 C8.20D20 C8.24D20 C40.9C23 D4.11D20 D4.13D20 SD16⋊D10 D8⋊6D10 D5×C8.C22 D20.44D4 Dic20⋊S3 C40.2D6 C60.63D4 C12.D20 C8.D30
C8.D10 is a maximal quotient of
C8⋊Dic10 C42.14D10 C42.16D10 C42.20D10 C8.D20 Dic20⋊9C4 C23.34D20 C23.10D20 D20.32D4 C22.D40 Dic10⋊14D4 C22⋊Dic20 Dic10.3Q8 C20⋊SD16 C42.36D10 D20⋊4Q8 C4⋊Dic20 Dic10⋊4Q8 C23.46D20 C23.47D20 C23.49D20 C40⋊2D4 C40.4D4 Dic20⋊S3 C40.2D6 C60.63D4 C12.D20 C8.D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 2 | 2 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D20 | D20 | C8.C22 | C8.D10 |
| kernel | C8.D10 | C40⋊C2 | Dic20 | C5×M4(2) | C2×Dic10 | C4○D20 | C20 | C2×C10 | M4(2) | C8 | C2×C4 | C4 | C22 | C5 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 1 | 4 |
Matrix representation of C8.D10 ►in GL4(𝔽41) generated by
| 30 | 29 | 28 | 24 |
| 0 | 1 | 13 | 28 |
| 10 | 35 | 40 | 12 |
| 9 | 10 | 0 | 11 |
| 10 | 14 | 7 | 31 |
| 24 | 33 | 9 | 7 |
| 13 | 32 | 8 | 27 |
| 0 | 13 | 17 | 31 |
| 39 | 9 | 29 | 21 |
| 10 | 34 | 21 | 9 |
| 5 | 9 | 9 | 34 |
| 12 | 9 | 36 | 0 |
G:=sub<GL(4,GF(41))| [30,0,10,9,29,1,35,10,28,13,40,0,24,28,12,11],[10,24,13,0,14,33,32,13,7,9,8,17,31,7,27,31],[39,10,5,12,9,34,9,9,29,21,9,36,21,9,34,0] >;
C8.D10 in GAP, Magma, Sage, TeX
C_8.D_{10} % in TeX
G:=Group("C8.D10"); // GroupNames label
G:=SmallGroup(160,130);
// by ID
G=gap.SmallGroup(160,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,188,50,579,69,4613]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^10=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^9>;
// generators/relations