metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊5D20, C42⋊15D10, C10.172+ (1+4), C4⋊C4⋊48D10, (C5×D4)⋊10D4, (C4×D4)⋊16D5, (C4×D20)⋊30C2, (D4×C20)⋊18C2, C20⋊7D4⋊9C2, C5⋊3(D4⋊5D4), C4.22(C2×D20), C20.54(C2×D4), D10⋊7(C4○D4), C22⋊D20⋊6C2, C4⋊2D20⋊15C2, (C4×C20)⋊20C22, C22⋊C4⋊47D10, (C2×D20)⋊6C22, (C22×C4)⋊13D10, C4⋊Dic5⋊8C22, C22.1(C2×D20), D10⋊2Q8⋊14C2, (C2×D4).248D10, C4.D20⋊18C2, (C2×C10).98C24, C2.18(C22×D20), C10.16(C22×D4), (C2×C20).159C23, (C22×C20)⋊10C22, C22.D20⋊5C2, C2.18(D4⋊6D10), D10⋊C4⋊52C22, (C2×Dic10)⋊17C22, (D4×C10).259C22, (C2×Dic5).42C23, (C22×Dic5)⋊9C22, (C22×D5).33C23, (C23×D5).40C22, C22.123(C23×D5), C23.172(C22×D5), (C22×C10).168C23, (C2×D4×D5)⋊4C2, (C2×C4×D5)⋊3C22, C2.22(D5×C4○D4), (C2×C10).1(C2×D4), (C2×D4⋊2D5)⋊3C2, (C5×C4⋊C4)⋊60C22, (C2×C5⋊D4)⋊4C22, C10.139(C2×C4○D4), (C2×D10⋊C4)⋊21C2, (C5×C22⋊C4)⋊50C22, (C2×C4).160(C22×D5), SmallGroup(320,1226)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1462 in 334 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×25], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×14], D5 [×5], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×19], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5 [×2], C22×D5 [×2], C22×D5 [×10], C22×C10 [×2], D4⋊5D4, C4⋊Dic5, C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×8], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×2], D4×D5 [×4], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×2], C4×D20, C4.D20, C22⋊D20 [×2], C22.D20 [×2], C4⋊2D20, D10⋊2Q8, C2×D10⋊C4 [×2], C20⋊7D4 [×2], D4×C20, C2×D4×D5, C2×D4⋊2D5, D4⋊5D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), D20 [×4], C22×D5 [×7], D4⋊5D4, C2×D20 [×6], C23×D5, C22×D20, D4⋊6D10, D5×C4○D4, D4⋊5D20
Generators and relations
G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 80 points
(1 49 32 71)(2 72 33 50)(3 51 34 73)(4 74 35 52)(5 53 36 75)(6 76 37 54)(7 55 38 77)(8 78 39 56)(9 57 40 79)(10 80 21 58)(11 59 22 61)(12 62 23 60)(13 41 24 63)(14 64 25 42)(15 43 26 65)(16 66 27 44)(17 45 28 67)(18 68 29 46)(19 47 30 69)(20 70 31 48)
(1 71)(2 50)(3 73)(4 52)(5 75)(6 54)(7 77)(8 56)(9 79)(10 58)(11 61)(12 60)(13 63)(14 42)(15 65)(16 44)(17 67)(18 46)(19 69)(20 48)(21 80)(22 59)(23 62)(24 41)(25 64)(26 43)(27 66)(28 45)(29 68)(30 47)(31 70)(32 49)(33 72)(34 51)(35 74)(36 53)(37 76)(38 55)(39 78)(40 57)
(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 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 59)(22 58)(23 57)(24 56)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 60)
G:=sub<Sym(80)| (1,49,32,71)(2,72,33,50)(3,51,34,73)(4,74,35,52)(5,53,36,75)(6,76,37,54)(7,55,38,77)(8,78,39,56)(9,57,40,79)(10,80,21,58)(11,59,22,61)(12,62,23,60)(13,41,24,63)(14,64,25,42)(15,43,26,65)(16,66,27,44)(17,45,28,67)(18,68,29,46)(19,47,30,69)(20,70,31,48), (1,71)(2,50)(3,73)(4,52)(5,75)(6,54)(7,77)(8,56)(9,79)(10,58)(11,61)(12,60)(13,63)(14,42)(15,65)(16,44)(17,67)(18,46)(19,69)(20,48)(21,80)(22,59)(23,62)(24,41)(25,64)(26,43)(27,66)(28,45)(29,68)(30,47)(31,70)(32,49)(33,72)(34,51)(35,74)(36,53)(37,76)(38,55)(39,78)(40,57), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,60)>;
G:=Group( (1,49,32,71)(2,72,33,50)(3,51,34,73)(4,74,35,52)(5,53,36,75)(6,76,37,54)(7,55,38,77)(8,78,39,56)(9,57,40,79)(10,80,21,58)(11,59,22,61)(12,62,23,60)(13,41,24,63)(14,64,25,42)(15,43,26,65)(16,66,27,44)(17,45,28,67)(18,68,29,46)(19,47,30,69)(20,70,31,48), (1,71)(2,50)(3,73)(4,52)(5,75)(6,54)(7,77)(8,56)(9,79)(10,58)(11,61)(12,60)(13,63)(14,42)(15,65)(16,44)(17,67)(18,46)(19,69)(20,48)(21,80)(22,59)(23,62)(24,41)(25,64)(26,43)(27,66)(28,45)(29,68)(30,47)(31,70)(32,49)(33,72)(34,51)(35,74)(36,53)(37,76)(38,55)(39,78)(40,57), (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), (1,70)(2,69)(3,68)(4,67)(5,66)(6,65)(7,64)(8,63)(9,62)(10,61)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,59)(22,58)(23,57)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(30,50)(31,49)(32,48)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,60) );
G=PermutationGroup([(1,49,32,71),(2,72,33,50),(3,51,34,73),(4,74,35,52),(5,53,36,75),(6,76,37,54),(7,55,38,77),(8,78,39,56),(9,57,40,79),(10,80,21,58),(11,59,22,61),(12,62,23,60),(13,41,24,63),(14,64,25,42),(15,43,26,65),(16,66,27,44),(17,45,28,67),(18,68,29,46),(19,47,30,69),(20,70,31,48)], [(1,71),(2,50),(3,73),(4,52),(5,75),(6,54),(7,77),(8,56),(9,79),(10,58),(11,61),(12,60),(13,63),(14,42),(15,65),(16,44),(17,67),(18,46),(19,69),(20,48),(21,80),(22,59),(23,62),(24,41),(25,64),(26,43),(27,66),(28,45),(29,68),(30,47),(31,70),(32,49),(33,72),(34,51),(35,74),(36,53),(37,76),(38,55),(39,78),(40,57)], [(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)], [(1,70),(2,69),(3,68),(4,67),(5,66),(6,65),(7,64),(8,63),(9,62),(10,61),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,59),(22,58),(23,57),(24,56),(25,55),(26,54),(27,53),(28,52),(29,51),(30,50),(31,49),(32,48),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,60)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 23 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 9 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 8 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 32 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 0 | 9 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,1,23,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,0,9],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,32,40,0,0,0,0,0,0,32,9,0,0,0,0,23,9] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | D20 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | D4⋊5D20 | C4×D20 | C4.D20 | C22⋊D20 | C22.D20 | C4⋊2D20 | D10⋊2Q8 | C2×D10⋊C4 | C20⋊7D4 | D4×C20 | C2×D4×D5 | C2×D4⋊2D5 | C5×D4 | C4×D4 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_4\rtimes_5D_{20} % in TeX
G:=Group("D4:5D20"); // GroupNames label
G:=SmallGroup(320,1226);
// by ID
G=gap.SmallGroup(320,1226);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations