metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.12D4, C23.4D10, C2.8(D4×D5), C4⋊Dic5⋊4C2, C22⋊C4⋊3D5, (C2×C4).6D10, C10.19(C2×D4), C23.D5⋊4C2, D10⋊C4⋊5C2, C10.8(C4○D4), C2.10(C4○D20), C10.D4⋊10C2, C2.8(D4⋊2D5), (C2×C20).52C22, (C2×C10).24C23, C5⋊1(C22.D4), (C2×Dic5).6C22, C22.42(C22×D5), (C22×C10).13C22, (C22×D5).20C22, (C2×C4×D5)⋊10C2, (C5×C22⋊C4)⋊5C2, (C2×C5⋊D4).3C2, SmallGroup(160,104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.12D4
G = < a,b,c,d | a10=b2=c4=1, d2=a5, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=a5c-1 >
Subgroups: 256 in 78 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C22.D4, C4×D5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, D10.12D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C22×D5, C4○D20, D4×D5, D4⋊2D5, D10.12D4
►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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
(1 11 25 78)(2 12 26 79)(3 13 27 80)(4 14 28 71)(5 15 29 72)(6 16 30 73)(7 17 21 74)(8 18 22 75)(9 19 23 76)(10 20 24 77)(31 68 48 51)(32 69 49 52)(33 70 50 53)(34 61 41 54)(35 62 42 55)(36 63 43 56)(37 64 44 57)(38 65 45 58)(39 66 46 59)(40 67 47 60)
(1 53 6 58)(2 54 7 59)(3 55 8 60)(4 56 9 51)(5 57 10 52)(11 45 16 50)(12 46 17 41)(13 47 18 42)(14 48 19 43)(15 49 20 44)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,11,25,78)(2,12,26,79)(3,13,27,80)(4,14,28,71)(5,15,29,72)(6,16,30,73)(7,17,21,74)(8,18,22,75)(9,19,23,76)(10,20,24,77)(31,68,48,51)(32,69,49,52)(33,70,50,53)(34,61,41,54)(35,62,42,55)(36,63,43,56)(37,64,44,57)(38,65,45,58)(39,66,46,59)(40,67,47,60), (1,53,6,58)(2,54,7,59)(3,55,8,60)(4,56,9,51)(5,57,10,52)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67), (1,11,25,78)(2,12,26,79)(3,13,27,80)(4,14,28,71)(5,15,29,72)(6,16,30,73)(7,17,21,74)(8,18,22,75)(9,19,23,76)(10,20,24,77)(31,68,48,51)(32,69,49,52)(33,70,50,53)(34,61,41,54)(35,62,42,55)(36,63,43,56)(37,64,44,57)(38,65,45,58)(39,66,46,59)(40,67,47,60), (1,53,6,58)(2,54,7,59)(3,55,8,60)(4,56,9,51)(5,57,10,52)(11,45,16,50)(12,46,17,41)(13,47,18,42)(14,48,19,43)(15,49,20,44)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,80),(19,79),(20,78),(31,41),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67)], [(1,11,25,78),(2,12,26,79),(3,13,27,80),(4,14,28,71),(5,15,29,72),(6,16,30,73),(7,17,21,74),(8,18,22,75),(9,19,23,76),(10,20,24,77),(31,68,48,51),(32,69,49,52),(33,70,50,53),(34,61,41,54),(35,62,42,55),(36,63,43,56),(37,64,44,57),(38,65,45,58),(39,66,46,59),(40,67,47,60)], [(1,53,6,58),(2,54,7,59),(3,55,8,60),(4,56,9,51),(5,57,10,52),(11,45,16,50),(12,46,17,41),(13,47,18,42),(14,48,19,43),(15,49,20,44),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)]])
D10.12D4 is a maximal subgroup of
C24.27D10 C24.30D10 C24.31D10 C42.93D10 C42.94D10 C42.95D10 C42.99D10 C42⋊12D10 D20⋊24D4 C42⋊16D10 C42.229D10 C42.114D10 C42.116D10 C42.118D10 C42.119D10 C24⋊3D10 C24.33D10 C24.35D10 C24.36D10 C10.342+ 1+4 C10.372+ 1+4 C4⋊C4⋊21D10 C10.732- 1+4 D20⋊20D4 C10.442+ 1+4 C10.472+ 1+4 C10.742- 1+4 C10.162- 1+4 D20⋊22D4 C10.202- 1+4 C10.212- 1+4 C10.242- 1+4 C10.582+ 1+4 C10.262- 1+4 D5×C22.D4 C10.822- 1+4 C10.1222+ 1+4 C10.622+ 1+4 C10.632+ 1+4 C10.642+ 1+4 C10.842- 1+4 C10.662+ 1+4 C10.852- 1+4 C10.682+ 1+4 C10.692+ 1+4 C42.137D10 C42.141D10 D20⋊10D4 C42⋊20D10 C42⋊21D10 C42.234D10 C42.143D10 C42.145D10 C42⋊23D10 C42⋊24D10 C42.189D10 C42.161D10 C42.162D10 C42.165D10 D30.35D4 D10.17D12 D30.7D4 D6⋊C4⋊D5 C23.17(S3×D5) D30.16D4 D30.28D4
D10.12D4 is a maximal quotient of
C10.51(C4×D4) C4⋊Dic5⋊15C4 C2.(C20⋊Q8) (C2×Dic5).Q8 C22.58(D4×D5) D10⋊3(C4⋊C4) (C2×C4).21D20 C10.(C4⋊D4) D10.12D8 D10.16SD16 C40⋊6C4⋊C2 C40⋊5C4⋊C2 D10.11SD16 D10.7Q16 (C2×C8).D10 D10⋊1C8.C2 C24.4D10 C24.7D10 C24.8D10 C24.9D10 C24.12D10 C24.14D10 C24.16D10 D30.35D4 D10.17D12 D30.7D4 D6⋊C4⋊D5 C23.17(S3×D5) D30.16D4 D30.28D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 2 | 2 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | C4○D20 | D4×D5 | D4⋊2D5 |
| kernel | D10.12D4 | C10.D4 | C4⋊Dic5 | D10⋊C4 | C23.D5 | C5×C22⋊C4 | C2×C4×D5 | C2×C5⋊D4 | D10 | C22⋊C4 | C10 | C2×C4 | C23 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 8 | 2 | 2 |
Matrix representation of D10.12D4 ►in GL4(𝔽41) generated by
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 34 | 34 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 30 | 32 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[34,1,0,0,34,7,0,0,0,0,1,0,0,0,0,40],[30,9,0,0,32,11,0,0,0,0,0,9,0,0,32,0],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,32] >;
D10.12D4 in GAP, Magma, Sage, TeX
D_{10}._{12}D_4 % in TeX
G:=Group("D10.12D4"); // GroupNames label
G:=SmallGroup(160,104);
// by ID
G=gap.SmallGroup(160,104);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,55,218,188,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=a^5*c^-1>;
// generators/relations