metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.8D10, C20.57D4, Q8.8D10, C20.18C23, D20.12C22, Dic10.11C22, D4⋊D5⋊7C2, C4○D4⋊2D5, C5⋊5(C4○D8), Q8⋊D5⋊7C2, C4○D20⋊4C2, D4.D5⋊7C2, C5⋊Q16⋊7C2, (C2×C10).9D4, (C2×C4).59D10, C10.60(C2×D4), C4.32(C5⋊D4), (C5×D4).8C22, C4.18(C22×D5), (C5×Q8).8C22, (C2×C20).43C22, C5⋊2C8.10C22, C22.1(C5⋊D4), (C2×C5⋊2C8)⋊8C2, (C5×C4○D4)⋊2C2, C2.24(C2×C5⋊D4), SmallGroup(160,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.8D10
G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c9 >
Subgroups: 184 in 62 conjugacy classes, 29 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4○D8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, D4.8D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C4○D8, C5⋊D4, C22×D5, C2×C5⋊D4, D4.8D10
►On 80 points
(1 70 11 80)(2 71 12 61)(3 72 13 62)(4 73 14 63)(5 74 15 64)(6 75 16 65)(7 76 17 66)(8 77 18 67)(9 78 19 68)(10 79 20 69)(21 45 31 55)(22 46 32 56)(23 47 33 57)(24 48 34 58)(25 49 35 59)(26 50 36 60)(27 51 37 41)(28 52 38 42)(29 53 39 43)(30 54 40 44)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)
(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 54 31 44)(22 43 32 53)(23 52 33 42)(24 41 34 51)(25 50 35 60)(26 59 36 49)(27 48 37 58)(28 57 38 47)(29 46 39 56)(30 55 40 45)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)
G:=sub<Sym(80)| (1,70,11,80)(2,71,12,61)(3,72,13,62)(4,73,14,63)(5,74,15,64)(6,75,16,65)(7,76,17,66)(8,77,18,67)(9,78,19,68)(10,79,20,69)(21,45,31,55)(22,46,32,56)(23,47,33,57)(24,48,34,58)(25,49,35,59)(26,50,36,60)(27,51,37,41)(28,52,38,42)(29,53,39,43)(30,54,40,44), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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)(21,54,31,44)(22,43,32,53)(23,52,33,42)(24,41,34,51)(25,50,35,60)(26,59,36,49)(27,48,37,58)(28,57,38,47)(29,46,39,56)(30,55,40,45)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)>;
G:=Group( (1,70,11,80)(2,71,12,61)(3,72,13,62)(4,73,14,63)(5,74,15,64)(6,75,16,65)(7,76,17,66)(8,77,18,67)(9,78,19,68)(10,79,20,69)(21,45,31,55)(22,46,32,56)(23,47,33,57)(24,48,34,58)(25,49,35,59)(26,50,36,60)(27,51,37,41)(28,52,38,42)(29,53,39,43)(30,54,40,44), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69), (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)(21,54,31,44)(22,43,32,53)(23,52,33,42)(24,41,34,51)(25,50,35,60)(26,59,36,49)(27,48,37,58)(28,57,38,47)(29,46,39,56)(30,55,40,45)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70) );
G=PermutationGroup([[(1,70,11,80),(2,71,12,61),(3,72,13,62),(4,73,14,63),(5,74,15,64),(6,75,16,65),(7,76,17,66),(8,77,18,67),(9,78,19,68),(10,79,20,69),(21,45,31,55),(22,46,32,56),(23,47,33,57),(24,48,34,58),(25,49,35,59),(26,50,36,60),(27,51,37,41),(28,52,38,42),(29,53,39,43),(30,54,40,44)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69)], [(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),(21,54,31,44),(22,43,32,53),(23,52,33,42),(24,41,34,51),(25,50,35,60),(26,59,36,49),(27,48,37,58),(28,57,38,47),(29,46,39,56),(30,55,40,45),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70)]])
D4.8D10 is a maximal subgroup of
M4(2).22D10 C42.196D10 C40.93D4 C40.50D4 D5×C4○D8 Q16⋊D10 D8⋊5D10 D8⋊6D10 C40.C23 D20.44D4 C20.C24 D20.32C23 D20.33C23 D20.34C23 D20.35C23 D20.34D6 C20.60D12 D20.24D6 C60.19C23 D20.27D6 Dic10.27D6 D4.8D30 C20.6S4
D4.8D10 is a maximal quotient of
C4⋊C4.233D10 C20.76(C4⋊C4) C4○D20⋊10C4 C4⋊C4.236D10 D4.3Dic10 C4×D4⋊D5 D4.1D20 C4×D4.D5 Q8.3Dic10 C4×Q8⋊D5 Q8.1D20 C4×C5⋊Q16 (C2×D4).D10 D20⋊17D4 (C2×C10)⋊D8 C5⋊2C8⋊23D4 C10.(C4○D8) D20.37D4 C5⋊2C8⋊24D4 (C2×C10)⋊Q16 C42.61D10 C42.213D10 D20.23D4 C42.214D10 Dic10.4Q8 C42.215D10 D20.4Q8 C42.216D10 C20.(C2×D4) (C5×D4)⋊14D4 (C5×D4).32D4 D20.34D6 C20.60D12 D20.24D6 C60.19C23 D20.27D6 Dic10.27D6 D4.8D30
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 20 | 1 | 1 | 2 | 4 | 20 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | D4.8D10 | C2×C5⋊2C8 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 | C4○D20 | C5×C4○D4 | C20 | C2×C10 | C4○D4 | C2×C4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of D4.8D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 18 |
| 0 | 0 | 9 | 40 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 26 | 0 |
| 7 | 7 | 0 | 0 |
| 34 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 7 | 7 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 1 | 9 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,9,0,0,18,40],[24,40,0,0,1,17,0,0,0,0,0,26,0,0,30,0],[7,34,0,0,7,40,0,0,0,0,32,0,0,0,0,32],[7,40,0,0,7,34,0,0,0,0,32,1,0,0,0,9] >;
D4.8D10 in GAP, Magma, Sage, TeX
D_4._8D_{10} % in TeX
G:=Group("D4.8D10"); // GroupNames label
G:=SmallGroup(160,171);
// by ID
G=gap.SmallGroup(160,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,218,579,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^9>;
// generators/relations