metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.37C24, D20.32C23, 2+ (1+4)⋊3D5, Dic10.32C23, C5⋊5(D4○D8), C4○D4⋊5D10, (C2×D4)⋊16D10, (C5×D4).36D4, (C5×Q8).36D4, D4⋊8D10⋊9C2, D4⋊D5⋊20C22, C20.269(C2×D4), Q8⋊D5⋊19C22, D4⋊D10⋊11C2, C4.37(C23×D5), D4.8D10⋊8C2, D4.18(C5⋊D4), C4○D20⋊10C22, (D4×C10)⋊24C22, (C2×D20)⋊39C22, Q8.Dic5⋊10C2, C5⋊2C8.16C23, D4.D5⋊19C22, Q8.18(C5⋊D4), D4.25(C22×D5), (C5×D4).25C23, C5⋊Q16⋊21C22, D4.D10⋊11C2, Q8.25(C22×D5), (C5×Q8).25C23, (C2×C20).118C23, C10.171(C22×D4), C4.Dic5⋊16C22, (C5×2+ (1+4))⋊2C2, (C2×D4⋊D5)⋊32C2, C4.75(C2×C5⋊D4), (C2×C10).85(C2×D4), (C5×C4○D4)⋊8C22, C22.6(C2×C5⋊D4), (C2×C5⋊2C8)⋊24C22, C2.44(C22×C5⋊D4), (C2×C4).102(C22×D5), SmallGroup(320,1507)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 982 in 268 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C8 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×6], D4 [×15], Q8 [×2], Q8, C23 [×6], D5 [×3], C10, C10 [×6], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×3], C2×D4 [×9], C4○D4, C4○D4 [×3], C4○D4 [×5], Dic5, C20, C20 [×3], C20, D10 [×6], C2×C10 [×3], C2×C10 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4), 2+ (1+4), C5⋊2C8, C5⋊2C8 [×3], Dic10, C4×D5 [×3], D20 [×3], D20 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×6], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×3], C22×C10 [×3], D4○D8, C2×C5⋊2C8 [×3], C4.Dic5 [×3], D4⋊D5 [×9], D4.D5 [×3], Q8⋊D5 [×3], C5⋊Q16, C2×D20 [×3], C4○D20 [×3], D4×D5 [×3], Q8⋊2D5, D4×C10 [×3], D4×C10 [×3], C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4, C2×D4⋊D5 [×3], D4.D10 [×3], Q8.Dic5, D4⋊D10 [×3], D4.8D10 [×3], D4⋊8D10, C5×2+ (1+4), D20.32C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], D4○D8, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.32C23
Generators and relations
G = < a,b,c,d,e | a20=b2=c2=d2=e2=1, bab=dad=a-1, ac=ca, eae=a11, cbc=a10b, dbd=a18b, ebe=a15b, cd=dc, ce=ec, ede=a5d >
►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 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)
(1 44)(2 43)(3 42)(4 41)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 80)(37 79)(38 78)(39 77)(40 76)
(1 36)(2 27)(3 38)(4 29)(5 40)(6 31)(7 22)(8 33)(9 24)(10 35)(11 26)(12 37)(13 28)(14 39)(15 30)(16 21)(17 32)(18 23)(19 34)(20 25)(41 62)(42 73)(43 64)(44 75)(45 66)(46 77)(47 68)(48 79)(49 70)(50 61)(51 72)(52 63)(53 74)(54 65)(55 76)(56 67)(57 78)(58 69)(59 80)(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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (1,44)(2,43)(3,42)(4,41)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,80)(37,79)(38,78)(39,77)(40,76), (1,36)(2,27)(3,38)(4,29)(5,40)(6,31)(7,22)(8,33)(9,24)(10,35)(11,26)(12,37)(13,28)(14,39)(15,30)(16,21)(17,32)(18,23)(19,34)(20,25)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(36,40)(37,39)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78), (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (1,44)(2,43)(3,42)(4,41)(5,60)(6,59)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,75)(22,74)(23,73)(24,72)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,80)(37,79)(38,78)(39,77)(40,76), (1,36)(2,27)(3,38)(4,29)(5,40)(6,31)(7,22)(8,33)(9,24)(10,35)(11,26)(12,37)(13,28)(14,39)(15,30)(16,21)(17,32)(18,23)(19,34)(20,25)(41,62)(42,73)(43,64)(44,75)(45,66)(46,77)(47,68)(48,79)(49,70)(50,61)(51,72)(52,63)(53,74)(54,65)(55,76)(56,67)(57,78)(58,69)(59,80)(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(36,40),(37,39),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78)], [(1,59),(2,60),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64)], [(1,44),(2,43),(3,42),(4,41),(5,60),(6,59),(7,58),(8,57),(9,56),(10,55),(11,54),(12,53),(13,52),(14,51),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,75),(22,74),(23,73),(24,72),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,80),(37,79),(38,78),(39,77),(40,76)], [(1,36),(2,27),(3,38),(4,29),(5,40),(6,31),(7,22),(8,33),(9,24),(10,35),(11,26),(12,37),(13,28),(14,39),(15,30),(16,21),(17,32),(18,23),(19,34),(20,25),(41,62),(42,73),(43,64),(44,75),(45,66),(46,77),(47,68),(48,79),(49,70),(50,61),(51,72),(52,63),(53,74),(54,65),(55,76),(56,67),(57,78),(58,69),(59,80),(60,71)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 1 | 0 | 0 | 0 | 0 |
| 33 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 33 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 24 |
| 0 | 0 | 0 | 0 | 29 | 17 |
| 0 | 0 | 24 | 24 | 0 | 0 |
| 0 | 0 | 29 | 17 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 12 | 24 |
| 0 | 0 | 24 | 24 | 0 | 0 |
| 0 | 0 | 29 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,17,0,0,24,29,0,0,0,0,24,17,0,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,0,0,24,29,0,0,0,0,24,0,0,0,0,12,0,0,0,0,17,24,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | ··· | 10T | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 2 | 2 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | C5⋊D4 | D4○D8 | D20.32C23 |
| kernel | D20.32C23 | C2×D4⋊D5 | D4.D10 | Q8.Dic5 | D4⋊D10 | D4.8D10 | D4⋊8D10 | C5×2+ (1+4) | C5×D4 | C5×Q8 | 2+ (1+4) | C2×D4 | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 3 | 1 | 2 | 6 | 8 | 12 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
D_{20}._{32}C_2^3 % in TeX
G:=Group("D20.32C2^3"); // GroupNames label
G:=SmallGroup(320,1507);
// by ID
G=gap.SmallGroup(320,1507);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=e^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e=a^11,c*b*c=a^10*b,d*b*d=a^18*b,e*b*e=a^15*b,c*d=d*c,c*e=e*c,e*d*e=a^5*d>;
// generators/relations