metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊10D4, C42⋊19D10, C10.1252+ (1+4), C4.70(D4×D5), (C4×D20)⋊43C2, C5⋊9(D4⋊5D4), (C2×Q8)⋊19D10, C20.63(C2×D4), C20⋊2D4⋊33C2, (C4×C20)⋊23C22, C22⋊C4⋊33D10, D10.47(C2×D4), C4.4D4⋊10D5, D10⋊11(C4○D4), C22⋊D20⋊24C2, D10⋊D4⋊40C2, D10⋊3Q8⋊28C2, (C2×D4).173D10, (C2×D20)⋊28C22, C4⋊Dic5⋊60C22, (Q8×C10)⋊13C22, C10.90(C22×D4), (C2×C10).220C24, (C2×C20).601C23, D10.12D4⋊42C2, C23.D5⋊33C22, C2.49(D4⋊8D10), D10⋊C4⋊54C22, C23.42(C22×D5), (D4×C10).155C22, C10.D4⋊26C22, (C22×C10).50C23, (C23×D5).64C22, C22.241(C23×D5), (C2×Dic5).115C23, (C22×D5).225C23, (C2×D4×D5)⋊17C2, C2.63(C2×D4×D5), C2.76(D5×C4○D4), (C2×C4×D5)⋊26C22, (D5×C22⋊C4)⋊17C2, (C2×Q8⋊2D5)⋊11C2, C10.187(C2×C4○D4), (C5×C4.4D4)⋊12C2, (C2×C5⋊D4)⋊23C22, (C5×C22⋊C4)⋊29C22, (C2×C4).195(C22×D5), SmallGroup(320,1348)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1478 in 334 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×29], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D5 [×7], C10 [×3], C10 [×2], C42, C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×6], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×6], D10 [×17], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D5 [×10], D20 [×4], D20 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], C22×D5 [×10], C22×C10 [×2], D4⋊5D4, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×4], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×2], C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×2], D4×D5 [×4], Q8⋊2D5 [×4], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C23×D5 [×2], C4×D20 [×2], D5×C22⋊C4 [×2], C22⋊D20 [×2], D10.12D4 [×2], D10⋊D4 [×2], C20⋊2D4, D10⋊3Q8, C5×C4.4D4, C2×D4×D5, C2×Q8⋊2D5, D20⋊10D4
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, D5×C4○D4, D4⋊8D10, D20⋊10D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a9, cbc-1=a10b, dbd=a18b, 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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)
(1 46 70 29)(2 47 71 30)(3 48 72 31)(4 49 73 32)(5 50 74 33)(6 51 75 34)(7 52 76 35)(8 53 77 36)(9 54 78 37)(10 55 79 38)(11 56 80 39)(12 57 61 40)(13 58 62 21)(14 59 63 22)(15 60 64 23)(16 41 65 24)(17 42 66 25)(18 43 67 26)(19 44 68 27)(20 45 69 28)
(1 56)(2 45)(3 54)(4 43)(5 52)(6 41)(7 50)(8 59)(9 48)(10 57)(11 46)(12 55)(13 44)(14 53)(15 42)(16 51)(17 60)(18 49)(19 58)(20 47)(21 68)(22 77)(23 66)(24 75)(25 64)(26 73)(27 62)(28 71)(29 80)(30 69)(31 78)(32 67)(33 76)(34 65)(35 74)(36 63)(37 72)(38 61)(39 70)(40 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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80), (1,46,70,29)(2,47,71,30)(3,48,72,31)(4,49,73,32)(5,50,74,33)(6,51,75,34)(7,52,76,35)(8,53,77,36)(9,54,78,37)(10,55,79,38)(11,56,80,39)(12,57,61,40)(13,58,62,21)(14,59,63,22)(15,60,64,23)(16,41,65,24)(17,42,66,25)(18,43,67,26)(19,44,68,27)(20,45,69,28), (1,56)(2,45)(3,54)(4,43)(5,52)(6,41)(7,50)(8,59)(9,48)(10,57)(11,46)(12,55)(13,44)(14,53)(15,42)(16,51)(17,60)(18,49)(19,58)(20,47)(21,68)(22,77)(23,66)(24,75)(25,64)(26,73)(27,62)(28,71)(29,80)(30,69)(31,78)(32,67)(33,76)(34,65)(35,74)(36,63)(37,72)(38,61)(39,70)(40,79)>;
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)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80), (1,46,70,29)(2,47,71,30)(3,48,72,31)(4,49,73,32)(5,50,74,33)(6,51,75,34)(7,52,76,35)(8,53,77,36)(9,54,78,37)(10,55,79,38)(11,56,80,39)(12,57,61,40)(13,58,62,21)(14,59,63,22)(15,60,64,23)(16,41,65,24)(17,42,66,25)(18,43,67,26)(19,44,68,27)(20,45,69,28), (1,56)(2,45)(3,54)(4,43)(5,52)(6,41)(7,50)(8,59)(9,48)(10,57)(11,46)(12,55)(13,44)(14,53)(15,42)(16,51)(17,60)(18,49)(19,58)(20,47)(21,68)(22,77)(23,66)(24,75)(25,64)(26,73)(27,62)(28,71)(29,80)(30,69)(31,78)(32,67)(33,76)(34,65)(35,74)(36,63)(37,72)(38,61)(39,70)(40,79) );
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),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80)], [(1,46,70,29),(2,47,71,30),(3,48,72,31),(4,49,73,32),(5,50,74,33),(6,51,75,34),(7,52,76,35),(8,53,77,36),(9,54,78,37),(10,55,79,38),(11,56,80,39),(12,57,61,40),(13,58,62,21),(14,59,63,22),(15,60,64,23),(16,41,65,24),(17,42,66,25),(18,43,67,26),(19,44,68,27),(20,45,69,28)], [(1,56),(2,45),(3,54),(4,43),(5,52),(6,41),(7,50),(8,59),(9,48),(10,57),(11,46),(12,55),(13,44),(14,53),(15,42),(16,51),(17,60),(18,49),(19,58),(20,47),(21,68),(22,77),(23,66),(24,75),(25,64),(26,73),(27,62),(28,71),(29,80),(30,69),(31,78),(32,67),(33,76),(34,65),(35,74),(36,63),(37,72),(38,61),(39,70),(40,79)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 7 | 1 | 0 | 0 | 0 | 0 |
| 33 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 40 |
| 0 | 0 | 0 | 0 | 11 | 25 |
| 40 | 40 | 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 | 16 | 40 |
| 0 | 0 | 0 | 0 | 9 | 25 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 0 | 0 | 0 | 0 | 24 | 21 |
| 7 | 6 | 0 | 0 | 0 | 0 |
| 33 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 32 |
| 0 | 0 | 0 | 0 | 17 | 20 |
G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,11,0,0,0,0,40,25],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,40,25],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,20,24,0,0,0,0,9,21],[7,33,0,0,0,0,6,34,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,21,17,0,0,0,0,32,20] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 2L | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| 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 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | ··· | 10 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | D5×C4○D4 | D4⋊8D10 |
| kernel | D20⋊10D4 | C4×D20 | D5×C22⋊C4 | C22⋊D20 | D10.12D4 | D10⋊D4 | C20⋊2D4 | D10⋊3Q8 | C5×C4.4D4 | C2×D4×D5 | C2×Q8⋊2D5 | D20 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 8 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{10}D_4 % in TeX
G:=Group("D20:10D4"); // GroupNames label
G:=SmallGroup(320,1348);
// by ID
G=gap.SmallGroup(320,1348);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,570,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,c*b*c^-1=a^10*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations