direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4○D8, D8⋊14D10, Q16⋊12D10, SD16⋊14D10, D40⋊18C22, C40.45C23, C20.14C24, D20.9C23, Dic20⋊16C22, Dic10.9C23, (D5×D8)⋊8C2, C4○D4⋊7D10, (C2×C8)⋊27D10, (D5×Q16)⋊8C2, D8⋊3D5⋊8C2, (C2×C40)⋊4C22, (D5×SD16)⋊7C2, C4.221(D4×D5), Q8.D10⋊8C2, D40⋊7C2⋊6C2, C22.4(D4×D5), D4⋊D5⋊12C22, (C4×D5).124D4, C20.380(C2×D4), C4○D20⋊5C22, (C5×D8)⋊12C22, (C8×D5)⋊16C22, Q8⋊D5⋊11C22, D4.8(C22×D5), (C5×D4).8C23, C4.14(C23×D5), C8.42(C22×D5), SD16⋊3D5⋊7C2, D10.114(C2×D4), D4.8D10⋊1C2, (C5×Q8).8C23, Q8.8(C22×D5), (Q8×D5).9C22, D4⋊2D5⋊8C22, C40⋊C2⋊20C22, C5⋊2C8.24C23, D4.D5⋊11C22, Q8⋊2D5⋊8C22, (C5×Q16)⋊10C22, (C22×D5).95D4, (C4×D5).65C23, C5⋊Q16⋊10C22, (D4×D5).10C22, (C2×C20).531C23, (C2×Dic5).169D4, Dic5.126(C2×D4), (C5×SD16)⋊15C22, C10.115(C22×D4), (D5×C2×C8)⋊1C2, C5⋊5(C2×C4○D8), C2.88(C2×D4×D5), (D5×C4○D4)⋊1C2, (C5×C4○D8)⋊2C2, (C2×C10).11(C2×D4), (C5×C4○D4)⋊1C22, (C2×C5⋊2C8)⋊37C22, (C2×C4×D5).331C22, (C2×C4).618(C22×D5), SmallGroup(320,1439)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4○D8
G = < a,b,c,d,e | a5=b2=c4=e2=1, d4=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d3 >
Subgroups: 1022 in 266 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, C2×C8, D8, D8, SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C2×C4○D8, C8×D5, C40⋊C2, D40, Dic20, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D5×C2×C8, D40⋊7C2, D5×D8, D8⋊3D5, D5×SD16, SD16⋊3D5, D5×Q16, Q8.D10, D4.8D10, C5×C4○D8, D5×C4○D4, D5×C4○D8
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C4○D8, C22×D4, C22×D5, C2×C4○D8, D4×D5, C23×D5, C2×D4×D5, D5×C4○D8
►On 80 points
(1 55 44 58 71)(2 56 45 59 72)(3 49 46 60 65)(4 50 47 61 66)(5 51 48 62 67)(6 52 41 63 68)(7 53 42 64 69)(8 54 43 57 70)(9 35 23 30 73)(10 36 24 31 74)(11 37 17 32 75)(12 38 18 25 76)(13 39 19 26 77)(14 40 20 27 78)(15 33 21 28 79)(16 34 22 29 80)
(1 71)(2 72)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 73)(24 74)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)
(1 40 5 36)(2 33 6 37)(3 34 7 38)(4 35 8 39)(9 70 13 66)(10 71 14 67)(11 72 15 68)(12 65 16 69)(17 56 21 52)(18 49 22 53)(19 50 23 54)(20 51 24 55)(25 46 29 42)(26 47 30 43)(27 48 31 44)(28 41 32 45)(57 77 61 73)(58 78 62 74)(59 79 63 75)(60 80 64 76)
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(33 39)(34 38)(35 37)(41 47)(42 46)(43 45)(49 53)(50 52)(54 56)(57 59)(60 64)(61 63)(65 69)(66 68)(70 72)(73 75)(76 80)(77 79)
G:=sub<Sym(80)| (1,55,44,58,71)(2,56,45,59,72)(3,49,46,60,65)(4,50,47,61,66)(5,51,48,62,67)(6,52,41,63,68)(7,53,42,64,69)(8,54,43,57,70)(9,35,23,30,73)(10,36,24,31,74)(11,37,17,32,75)(12,38,18,25,76)(13,39,19,26,77)(14,40,20,27,78)(15,33,21,28,79)(16,34,22,29,80), (1,71)(2,72)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,73)(24,74)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,40,5,36)(2,33,6,37)(3,34,7,38)(4,35,8,39)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(17,56,21,52)(18,49,22,53)(19,50,23,54)(20,51,24,55)(25,46,29,42)(26,47,30,43)(27,48,31,44)(28,41,32,45)(57,77,61,73)(58,78,62,74)(59,79,63,75)(60,80,64,76), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,23)(18,22)(19,21)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45)(49,53)(50,52)(54,56)(57,59)(60,64)(61,63)(65,69)(66,68)(70,72)(73,75)(76,80)(77,79)>;
G:=Group( (1,55,44,58,71)(2,56,45,59,72)(3,49,46,60,65)(4,50,47,61,66)(5,51,48,62,67)(6,52,41,63,68)(7,53,42,64,69)(8,54,43,57,70)(9,35,23,30,73)(10,36,24,31,74)(11,37,17,32,75)(12,38,18,25,76)(13,39,19,26,77)(14,40,20,27,78)(15,33,21,28,79)(16,34,22,29,80), (1,71)(2,72)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,73)(24,74)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59), (1,40,5,36)(2,33,6,37)(3,34,7,38)(4,35,8,39)(9,70,13,66)(10,71,14,67)(11,72,15,68)(12,65,16,69)(17,56,21,52)(18,49,22,53)(19,50,23,54)(20,51,24,55)(25,46,29,42)(26,47,30,43)(27,48,31,44)(28,41,32,45)(57,77,61,73)(58,78,62,74)(59,79,63,75)(60,80,64,76), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,23)(18,22)(19,21)(25,29)(26,28)(30,32)(33,39)(34,38)(35,37)(41,47)(42,46)(43,45)(49,53)(50,52)(54,56)(57,59)(60,64)(61,63)(65,69)(66,68)(70,72)(73,75)(76,80)(77,79) );
G=PermutationGroup([[(1,55,44,58,71),(2,56,45,59,72),(3,49,46,60,65),(4,50,47,61,66),(5,51,48,62,67),(6,52,41,63,68),(7,53,42,64,69),(8,54,43,57,70),(9,35,23,30,73),(10,36,24,31,74),(11,37,17,32,75),(12,38,18,25,76),(13,39,19,26,77),(14,40,20,27,78),(15,33,21,28,79),(16,34,22,29,80)], [(1,71),(2,72),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,33),(16,34),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,73),(24,74),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59)], [(1,40,5,36),(2,33,6,37),(3,34,7,38),(4,35,8,39),(9,70,13,66),(10,71,14,67),(11,72,15,68),(12,65,16,69),(17,56,21,52),(18,49,22,53),(19,50,23,54),(20,51,24,55),(25,46,29,42),(26,47,30,43),(27,48,31,44),(28,41,32,45),(57,77,61,73),(58,78,62,74),(59,79,63,75),(60,80,64,76)], [(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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,23),(18,22),(19,21),(25,29),(26,28),(30,32),(33,39),(34,38),(35,37),(41,47),(42,46),(43,45),(49,53),(50,52),(54,56),(57,59),(60,64),(61,63),(65,69),(66,68),(70,72),(73,75),(76,80),(77,79)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 20 | 20 | 1 | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | C4○D8 | D4×D5 | D4×D5 | D5×C4○D8 |
| kernel | D5×C4○D8 | D5×C2×C8 | D40⋊7C2 | D5×D8 | D8⋊3D5 | D5×SD16 | SD16⋊3D5 | D5×Q16 | Q8.D10 | D4.8D10 | C5×C4○D8 | D5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | D5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of D5×C4○D8 ►in GL4(𝔽41) generated by
| 7 | 1 | 0 | 0 |
| 33 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [7,33,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,29,12],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;
D5×C4○D8 in GAP, Magma, Sage, TeX
D_5\times C_4\circ D_8
% in TeX
G:=Group("D5xC4oD8"); // GroupNames label
G:=SmallGroup(320,1439);
// by ID
G=gap.SmallGroup(320,1439);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^2=1,d^4=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^3>;
// generators/relations