metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊21D4, C10.1172+ 1+4, C4⋊C4⋊10D10, C22⋊Q8⋊7D5, C5⋊7(D4⋊5D4), (C2×Q8)⋊16D10, C4.111(D4×D5), C4⋊D20⋊25C2, D10.43(C2×D4), C20.234(C2×D4), D20⋊8C4⋊25C2, C22⋊D20⋊16C2, (Q8×C10)⋊7C22, (C22×D20)⋊16C2, (C2×D20)⋊25C22, (C2×C20).54C23, C22⋊C4.57D10, C10.76(C22×D4), C20.23D4⋊12C2, (C2×C10).174C24, C22⋊2(Q8⋊2D5), (C4×Dic5)⋊28C22, (C22×C4).236D10, D10.13D4⋊17C2, C2.34(D4⋊8D10), D10⋊C4⋊20C22, C10.D4⋊53C22, (C23×D5).52C22, C23.189(C22×D5), C22.195(C23×D5), (C22×C10).202C23, (C22×C20).254C22, (C2×Dic5).243C23, (C22×D5).206C23, C23.D5.115C22, C2.49(C2×D4×D5), (C4×C5⋊D4)⋊22C2, (D5×C22⋊C4)⋊8C2, (C2×C4×D5)⋊18C22, (C2×C10)⋊7(C4○D4), (C5×C4⋊C4)⋊19C22, (C2×Q8⋊2D5)⋊7C2, (C5×C22⋊Q8)⋊10C2, C10.114(C2×C4○D4), C2.17(C2×Q8⋊2D5), (C2×C4).47(C22×D5), (C2×C5⋊D4).130C22, (C5×C22⋊C4).29C22, SmallGroup(320,1302)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊21D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=dbd=a18b, dcd=c-1 >
Subgroups: 1510 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×D5, C22×D5, C22×C10, D4⋊5D4, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×D20, Q8⋊2D5, C2×C5⋊D4, C22×C20, Q8×C10, C23×D5, D5×C22⋊C4, C22⋊D20, D20⋊8C4, D10.13D4, C4⋊D20, C4⋊D20, C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C22×D20, C2×Q8⋊2D5, D20⋊21D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C22×D5, D4⋊5D4, D4×D5, Q8⋊2D5, C23×D5, C2×D4×D5, C2×Q8⋊2D5, D4⋊8D10, D20⋊21D4
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(76 80)(77 79)
(1 32 61 43)(2 21 62 52)(3 30 63 41)(4 39 64 50)(5 28 65 59)(6 37 66 48)(7 26 67 57)(8 35 68 46)(9 24 69 55)(10 33 70 44)(11 22 71 53)(12 31 72 42)(13 40 73 51)(14 29 74 60)(15 38 75 49)(16 27 76 58)(17 36 77 47)(18 25 78 56)(19 34 79 45)(20 23 80 54)
(1 53)(2 42)(3 51)(4 60)(5 49)(6 58)(7 47)(8 56)(9 45)(10 54)(11 43)(12 52)(13 41)(14 50)(15 59)(16 48)(17 57)(18 46)(19 55)(20 44)(21 72)(22 61)(23 70)(24 79)(25 68)(26 77)(27 66)(28 75)(29 64)(30 73)(31 62)(32 71)(33 80)(34 69)(35 78)(36 67)(37 76)(38 65)(39 74)(40 63)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79), (1,32,61,43)(2,21,62,52)(3,30,63,41)(4,39,64,50)(5,28,65,59)(6,37,66,48)(7,26,67,57)(8,35,68,46)(9,24,69,55)(10,33,70,44)(11,22,71,53)(12,31,72,42)(13,40,73,51)(14,29,74,60)(15,38,75,49)(16,27,76,58)(17,36,77,47)(18,25,78,56)(19,34,79,45)(20,23,80,54), (1,53)(2,42)(3,51)(4,60)(5,49)(6,58)(7,47)(8,56)(9,45)(10,54)(11,43)(12,52)(13,41)(14,50)(15,59)(16,48)(17,57)(18,46)(19,55)(20,44)(21,72)(22,61)(23,70)(24,79)(25,68)(26,77)(27,66)(28,75)(29,64)(30,73)(31,62)(32,71)(33,80)(34,69)(35,78)(36,67)(37,76)(38,65)(39,74)(40,63)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(76,80)(77,79), (1,32,61,43)(2,21,62,52)(3,30,63,41)(4,39,64,50)(5,28,65,59)(6,37,66,48)(7,26,67,57)(8,35,68,46)(9,24,69,55)(10,33,70,44)(11,22,71,53)(12,31,72,42)(13,40,73,51)(14,29,74,60)(15,38,75,49)(16,27,76,58)(17,36,77,47)(18,25,78,56)(19,34,79,45)(20,23,80,54), (1,53)(2,42)(3,51)(4,60)(5,49)(6,58)(7,47)(8,56)(9,45)(10,54)(11,43)(12,52)(13,41)(14,50)(15,59)(16,48)(17,57)(18,46)(19,55)(20,44)(21,72)(22,61)(23,70)(24,79)(25,68)(26,77)(27,66)(28,75)(29,64)(30,73)(31,62)(32,71)(33,80)(34,69)(35,78)(36,67)(37,76)(38,65)(39,74)(40,63) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,27),(22,26),(23,25),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(76,80),(77,79)], [(1,32,61,43),(2,21,62,52),(3,30,63,41),(4,39,64,50),(5,28,65,59),(6,37,66,48),(7,26,67,57),(8,35,68,46),(9,24,69,55),(10,33,70,44),(11,22,71,53),(12,31,72,42),(13,40,73,51),(14,29,74,60),(15,38,75,49),(16,27,76,58),(17,36,77,47),(18,25,78,56),(19,34,79,45),(20,23,80,54)], [(1,53),(2,42),(3,51),(4,60),(5,49),(6,58),(7,47),(8,56),(9,45),(10,54),(11,43),(12,52),(13,41),(14,50),(15,59),(16,48),(17,57),(18,46),(19,55),(20,44),(21,72),(22,61),(23,70),(24,79),(25,68),(26,77),(27,66),(28,75),(29,64),(30,73),(31,62),(32,71),(33,80),(34,69),(35,78),(36,67),(37,76),(38,65),(39,74),(40,63)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 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 | Q8⋊2D5 | D4⋊8D10 |
| kernel | D20⋊21D4 | D5×C22⋊C4 | C22⋊D20 | D20⋊8C4 | D10.13D4 | C4⋊D20 | C4×C5⋊D4 | C20.23D4 | C5×C22⋊Q8 | C22×D20 | C2×Q8⋊2D5 | D20 | C22⋊Q8 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of D20⋊21D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 36 |
| 0 | 0 | 0 | 0 | 25 | 1 |
| 0 | 1 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 25 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 0 | 0 | 0 | 0 | 21 | 32 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,25,0,0,0,0,36,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,25,0,0,0,0,0,1],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,20,0,0,0,0,37,9],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,21,0,0,0,0,4,32] >;
D20⋊21D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{21}D_4 % in TeX
G:=Group("D20:21D4"); // GroupNames label
G:=SmallGroup(320,1302);
// by ID
G=gap.SmallGroup(320,1302);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,192,12550]);
// 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^9,c*b*c^-1=d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations