metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.18C25, C20.53C24, D20.39C23, D10.10C24, 2- 1+4⋊5D5, Dic5.13C24, Dic10.40C23, C4○D4⋊13D10, (C2×Q8)⋊25D10, (D4×D5)⋊15C22, (C2×C10).9C24, D4⋊8D10⋊12C2, (Q8×D5)⋊18C22, C2.19(D5×C24), C4.50(C23×D5), C5⋊D4.5C23, C4○D20⋊15C22, (C2×D20)⋊41C22, C5⋊3(C2.C25), (Q8×C10)⋊25C22, D4.33(C22×D5), (C5×D4).33C23, (C4×D5).22C23, (C5×Q8).34C23, Q8.34(C22×D5), D4⋊2D5⋊19C22, C22.6(C23×D5), (C2×C20).124C23, Q8.10D10⋊8C2, Q8⋊2D5⋊17C22, (C5×2- 1+4)⋊5C2, (C2×Dic5).310C23, (C22×D5).144C23, (D5×C4○D4)⋊10C2, (C2×C4×D5)⋊38C22, (C2×Q8⋊2D5)⋊22C2, (C5×C4○D4)⋊13C22, (C2×C4).108(C22×D5), SmallGroup(320,1625)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.39C23
G = < a,b,c,d,e | a20=b2=c2=d2=e2=1, bab=a-1, ac=ca, ad=da, eae=a9, cbc=a10b, bd=db, ebe=a18b, dcd=ece=a10c, de=ed >
Subgroups: 2478 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C4○D4, 2+ 1+4, 2- 1+4, 2- 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2.C25, C2×C4×D5, C2×D20, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, Q8×C10, C5×C4○D4, C2×Q8⋊2D5, Q8.10D10, D5×C4○D4, D4⋊8D10, C5×2- 1+4, D20.39C23
Quotients: C1, C2, C22, C23, D5, C24, D10, C25, C22×D5, C2.C25, C23×D5, D5×C24, D20.39C23
►On 80 points
Generators in S80
(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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 32)(2 21)(3 30)(4 39)(5 28)(6 37)(7 26)(8 35)(9 24)(10 33)(11 22)(12 31)(13 40)(14 29)(15 38)(16 27)(17 36)(18 25)(19 34)(20 23)(41 80)(42 69)(43 78)(44 67)(45 76)(46 65)(47 74)(48 63)(49 72)(50 61)(51 70)(52 79)(53 68)(54 77)(55 66)(56 75)(57 64)(58 73)(59 62)(60 71)
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,32)(2,21)(3,30)(4,39)(5,28)(6,37)(7,26)(8,35)(9,24)(10,33)(11,22)(12,31)(13,40)(14,29)(15,38)(16,27)(17,36)(18,25)(19,34)(20,23)(41,80)(42,69)(43,78)(44,67)(45,76)(46,65)(47,74)(48,63)(49,72)(50,61)(51,70)(52,79)(53,68)(54,77)(55,66)(56,75)(57,64)(58,73)(59,62)(60,71)>;
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,79)(18,80)(19,61)(20,62)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,32)(2,21)(3,30)(4,39)(5,28)(6,37)(7,26)(8,35)(9,24)(10,33)(11,22)(12,31)(13,40)(14,29)(15,38)(16,27)(17,36)(18,25)(19,34)(20,23)(41,80)(42,69)(43,78)(44,67)(45,76)(46,65)(47,74)(48,63)(49,72)(50,61)(51,70)(52,79)(53,68)(54,77)(55,66)(56,75)(57,64)(58,73)(59,62)(60,71) );
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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,79),(18,80),(19,61),(20,62),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,32),(2,21),(3,30),(4,39),(5,28),(6,37),(7,26),(8,35),(9,24),(10,33),(11,22),(12,31),(13,40),(14,29),(15,38),(16,27),(17,36),(18,25),(19,34),(20,23),(41,80),(42,69),(43,78),(44,67),(45,76),(46,65),(47,74),(48,63),(49,72),(50,61),(51,70),(52,79),(53,68),(54,77),(55,66),(56,75),(57,64),(58,73),(59,62),(60,71)]])
68 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | ··· | 2P | 4A | ··· | 4J | 4K | 4L | 4M | ··· | 4Q | 5A | 5B | 10A | 10B | 10C | ··· | 10L | 20A | ··· | 20T |
| order | 1 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | C2.C25 | D20.39C23 |
| kernel | D20.39C23 | C2×Q8⋊2D5 | Q8.10D10 | D5×C4○D4 | D4⋊8D10 | C5×2- 1+4 | 2- 1+4 | C2×Q8 | C4○D4 | C5 | C1 |
| # reps | 1 | 5 | 5 | 10 | 10 | 1 | 2 | 10 | 20 | 2 | 2 |
Matrix representation of D20.39C23 ►in GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 20 | 4 | 2 |
| 0 | 0 | 4 | 1 | 0 | 37 |
| 0 | 0 | 0 | 0 | 40 | 20 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 40 | 20 |
| 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 20 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 2 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 39 | 0 | 1 | 2 |
| 0 | 0 | 0 | 39 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 25 | 27 | 16 |
| 0 | 0 | 5 | 32 | 36 | 4 |
| 0 | 0 | 0 | 0 | 32 | 16 |
| 0 | 0 | 0 | 0 | 36 | 9 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,4,0,0,0,0,20,1,0,0,0,0,4,0,40,4,0,0,2,37,20,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,40,0,40,0,0,0,20,1,20,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,39,0,0,0,39,40,0,39,0,0,0,0,1,0,0,0,2,0,2,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,40,0,40,0,0,0,39,40,0,40],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,9,5,0,0,0,0,25,32,0,0,0,0,27,36,32,36,0,0,16,4,16,9] >;
D20.39C23 in GAP, Magma, Sage, TeX
D_{20}._{39}C_2^3 % in TeX
G:=Group("D20.39C2^3"); // GroupNames label
G:=SmallGroup(320,1625);
// by ID
G=gap.SmallGroup(320,1625);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^9,c*b*c=a^10*b,b*d=d*b,e*b*e=a^18*b,d*c*d=e*c*e=a^10*c,d*e=e*d>;
// generators/relations