metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊20D4, C10.412+ (1+4), C4⋊C4⋊24D10, (C2×D4)⋊8D10, C4⋊D4⋊15D5, C5⋊6(D4⋊5D4), C4.110(D4×D5), D10⋊9(C4○D4), C20⋊2D4⋊21C2, C22⋊C4⋊28D10, D10.42(C2×D4), C20.229(C2×D4), (C22×C4)⋊19D10, D20⋊8C4⋊21C2, C23⋊D10⋊11C2, D10⋊2Q8⋊22C2, (D4×C10)⋊14C22, C4⋊Dic5⋊32C22, C10.71(C22×D4), C20.17D4⋊17C2, (C2×C10).156C24, (C2×C20).595C23, (C22×C20)⋊22C22, (C4×Dic5)⋊23C22, D10.12D4⋊20C2, C2.43(D4⋊6D10), C23.D5⋊24C22, D10⋊C4⋊18C22, (C2×Dic10)⋊62C22, (C2×D20).272C22, C10.D4⋊65C22, (C22×C10).23C23, (C2×Dic5).75C23, (C23×D5).48C22, C22.177(C23×D5), C23.113(C22×D5), (C22×D5).200C23, (C2×D4×D5)⋊13C2, C2.44(C2×D4×D5), (C4×C5⋊D4)⋊18C2, (D5×C22⋊C4)⋊6C2, C2.40(D5×C4○D4), (C2×C4×D5)⋊15C22, (C2×C4○D20)⋊22C2, (C5×C4⋊D4)⋊18C2, (C5×C4⋊C4)⋊13C22, C10.153(C2×C4○D4), (C2×C5⋊D4)⋊40C22, (C2×C4).39(C22×D5), (C5×C22⋊C4)⋊15C22, SmallGroup(320,1284)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1414 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×18], Q8 [×2], C23, C23 [×2], C23 [×13], D5 [×6], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×5], C20 [×2], C20 [×3], D10 [×6], D10 [×14], C2×C10, C2×C10 [×9], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×8], D20 [×4], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×10], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×D5 [×10], C22×C10, C22×C10 [×2], D4⋊5D4, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4, D10⋊C4 [×4], C23.D5, C23.D5 [×4], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5 [×4], C2×D20, C4○D20 [×4], D4×D5 [×4], C2×C5⋊D4, C2×C5⋊D4 [×4], C22×C20, D4×C10, D4×C10 [×2], C23×D5 [×2], D5×C22⋊C4 [×2], D10.12D4 [×2], D20⋊8C4, D10⋊2Q8, C4×C5⋊D4, C20.17D4, C23⋊D10 [×2], C20⋊2D4 [×2], C5×C4⋊D4, C2×C4○D20, C2×D4×D5, D20⋊20D4
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), C22×D5 [×7], D4⋊5D4, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10, D5×C4○D4, D20⋊20D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, bc=cb, dbd=a10b, dcd=c-1 >
►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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 30)(39 40)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)
(1 25 51 80)(2 36 52 71)(3 27 53 62)(4 38 54 73)(5 29 55 64)(6 40 56 75)(7 31 57 66)(8 22 58 77)(9 33 59 68)(10 24 60 79)(11 35 41 70)(12 26 42 61)(13 37 43 72)(14 28 44 63)(15 39 45 74)(16 30 46 65)(17 21 47 76)(18 32 48 67)(19 23 49 78)(20 34 50 69)
(1 51)(2 42)(3 53)(4 44)(5 55)(6 46)(7 57)(8 48)(9 59)(10 50)(11 41)(12 52)(13 43)(14 54)(15 45)(16 56)(17 47)(18 58)(19 49)(20 60)(22 32)(24 34)(26 36)(28 38)(30 40)(61 71)(63 73)(65 75)(67 77)(69 79)
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75), (1,25,51,80)(2,36,52,71)(3,27,53,62)(4,38,54,73)(5,29,55,64)(6,40,56,75)(7,31,57,66)(8,22,58,77)(9,33,59,68)(10,24,60,79)(11,35,41,70)(12,26,42,61)(13,37,43,72)(14,28,44,63)(15,39,45,74)(16,30,46,65)(17,21,47,76)(18,32,48,67)(19,23,49,78)(20,34,50,69), (1,51)(2,42)(3,53)(4,44)(5,55)(6,46)(7,57)(8,48)(9,59)(10,50)(11,41)(12,52)(13,43)(14,54)(15,45)(16,56)(17,47)(18,58)(19,49)(20,60)(22,32)(24,34)(26,36)(28,38)(30,40)(61,71)(63,73)(65,75)(67,77)(69,79)>;
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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(39,40)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75), (1,25,51,80)(2,36,52,71)(3,27,53,62)(4,38,54,73)(5,29,55,64)(6,40,56,75)(7,31,57,66)(8,22,58,77)(9,33,59,68)(10,24,60,79)(11,35,41,70)(12,26,42,61)(13,37,43,72)(14,28,44,63)(15,39,45,74)(16,30,46,65)(17,21,47,76)(18,32,48,67)(19,23,49,78)(20,34,50,69), (1,51)(2,42)(3,53)(4,44)(5,55)(6,46)(7,57)(8,48)(9,59)(10,50)(11,41)(12,52)(13,43)(14,54)(15,45)(16,56)(17,47)(18,58)(19,49)(20,60)(22,32)(24,34)(26,36)(28,38)(30,40)(61,71)(63,73)(65,75)(67,77)(69,79) );
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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,30),(39,40),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75)], [(1,25,51,80),(2,36,52,71),(3,27,53,62),(4,38,54,73),(5,29,55,64),(6,40,56,75),(7,31,57,66),(8,22,58,77),(9,33,59,68),(10,24,60,79),(11,35,41,70),(12,26,42,61),(13,37,43,72),(14,28,44,63),(15,39,45,74),(16,30,46,65),(17,21,47,76),(18,32,48,67),(19,23,49,78),(20,34,50,69)], [(1,51),(2,42),(3,53),(4,44),(5,55),(6,46),(7,57),(8,48),(9,59),(10,50),(11,41),(12,52),(13,43),(14,54),(15,45),(16,56),(17,47),(18,58),(19,49),(20,60),(22,32),(24,34),(26,36),(28,38),(30,40),(61,71),(63,73),(65,75),(67,77),(69,79)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 32 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 40 | 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 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,32,0,0,0,0,32,0,0,0,0,0,0,0,7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | ··· | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 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 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2+ (1+4) | D4×D5 | D4⋊6D10 | D5×C4○D4 |
| kernel | D20⋊20D4 | D5×C22⋊C4 | D10.12D4 | D20⋊8C4 | D10⋊2Q8 | C4×C5⋊D4 | C20.17D4 | C23⋊D10 | C20⋊2D4 | C5×C4⋊D4 | C2×C4○D20 | C2×D4×D5 | D20 | C4⋊D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{20}D_4 % in TeX
G:=Group("D20:20D4"); // GroupNames label
G:=SmallGroup(320,1284);
// by ID
G=gap.SmallGroup(320,1284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations