metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.53C24, C10.18C25, D10.10C24, D20.39C23, 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, C4.50(C23×D5), C2.19(D5×C24), C5⋊D4.5C23, C4○D20⋊15C22, (C2×D20)⋊41C22, C5⋊3(C2.C25), (Q8×C10)⋊25C22, (C4×D5).22C23, (C5×D4).33C23, D4.33(C22×D5), (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
Subgroups: 2478 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C4 [×10], C4 [×6], C22 [×5], C22 [×25], C5, C2×C4 [×15], C2×C4 [×45], D4 [×10], D4 [×50], Q8 [×10], Q8 [×10], C23 [×15], D5 [×10], C10, C10 [×5], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×5], C2×Q8 [×10], C4○D4 [×10], C4○D4 [×70], Dic5, Dic5 [×5], C20 [×10], D10 [×10], D10 [×15], C2×C10 [×5], C2×C4○D4 [×15], 2+ (1+4) [×10], 2- (1+4), 2- (1+4) [×5], Dic10 [×10], C4×D5 [×40], D20 [×30], C2×Dic5 [×5], C5⋊D4 [×20], C2×C20 [×15], C5×D4 [×10], C5×Q8 [×10], C22×D5 [×15], C2.C25, C2×C4×D5 [×15], C2×D20 [×15], C4○D20 [×30], D4×D5 [×30], D4⋊2D5 [×10], Q8×D5 [×10], Q8⋊2D5 [×30], Q8×C10 [×5], C5×C4○D4 [×10], C2×Q8⋊2D5 [×5], Q8.10D10 [×5], D5×C4○D4 [×10], D4⋊8D10 [×10], C5×2- (1+4), D20.39C23
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], C25, C22×D5 [×35], C2.C25, C23×D5 [×15], D5×C24, D20.39C23
Generators and relations
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 >
►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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 49)(42 48)(43 47)(44 46)(50 60)(51 59)(52 58)(53 57)(54 56)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 58)(2 47)(3 56)(4 45)(5 54)(6 43)(7 52)(8 41)(9 50)(10 59)(11 48)(12 57)(13 46)(14 55)(15 44)(16 53)(17 42)(18 51)(19 60)(20 49)(21 69)(22 78)(23 67)(24 76)(25 65)(26 74)(27 63)(28 72)(29 61)(30 70)(31 79)(32 68)(33 77)(34 66)(35 75)(36 64)(37 73)(38 62)(39 71)(40 80)
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)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,58)(2,47)(3,56)(4,45)(5,54)(6,43)(7,52)(8,41)(9,50)(10,59)(11,48)(12,57)(13,46)(14,55)(15,44)(16,53)(17,42)(18,51)(19,60)(20,49)(21,69)(22,78)(23,67)(24,76)(25,65)(26,74)(27,63)(28,72)(29,61)(30,70)(31,79)(32,68)(33,77)(34,66)(35,75)(36,64)(37,73)(38,62)(39,71)(40,80)>;
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)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(34,40)(35,39)(36,38)(41,49)(42,48)(43,47)(44,46)(50,60)(51,59)(52,58)(53,57)(54,56)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52)(40,53), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,58)(2,47)(3,56)(4,45)(5,54)(6,43)(7,52)(8,41)(9,50)(10,59)(11,48)(12,57)(13,46)(14,55)(15,44)(16,53)(17,42)(18,51)(19,60)(20,49)(21,69)(22,78)(23,67)(24,76)(25,65)(26,74)(27,63)(28,72)(29,61)(30,70)(31,79)(32,68)(33,77)(34,66)(35,75)(36,64)(37,73)(38,62)(39,71)(40,80) );
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),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(34,40),(35,39),(36,38),(41,49),(42,48),(43,47),(44,46),(50,60),(51,59),(52,58),(53,57),(54,56),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52),(40,53)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,58),(2,47),(3,56),(4,45),(5,54),(6,43),(7,52),(8,41),(9,50),(10,59),(11,48),(12,57),(13,46),(14,55),(15,44),(16,53),(17,42),(18,51),(19,60),(20,49),(21,69),(22,78),(23,67),(24,76),(25,65),(26,74),(27,63),(28,72),(29,61),(30,70),(31,79),(32,68),(33,77),(34,66),(35,75),(36,64),(37,73),(38,62),(39,71),(40,80)])
Matrix representation ►G ⊆ 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] >;
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 |
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