metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.38C24, D20.33C23, 2+ (1+4)⋊4D5, Dic10.33C23, (C5×D4).37D4, C5⋊6(D4○SD16), (C5×Q8).37D4, D4⋊D5⋊21C22, C4○D4.16D10, C20.270(C2×D4), Q8⋊D5⋊22C22, C4.38(C23×D5), D4.8D10⋊9C2, (C2×D4).118D10, D4.19(C5⋊D4), C4○D20⋊11C22, Q8.Dic5⋊11C2, C5⋊2C8.17C23, D4.D5⋊21C22, Q8.19(C5⋊D4), D4.26(C22×D5), C5⋊Q16⋊18C22, (C5×D4).26C23, D4.D10⋊12C2, D4.10D10⋊9C2, D4.9D10⋊11C2, (C5×Q8).26C23, Q8.26(C22×D5), (C2×C20).119C23, C10.172(C22×D4), C4.Dic5⋊17C22, (C5×2+ (1+4))⋊3C2, (C2×Dic10)⋊43C22, (D4×C10).169C22, C4.76(C2×C5⋊D4), (C2×D4.D5)⋊32C2, (C2×C10).86(C2×D4), C22.7(C2×C5⋊D4), (C2×C5⋊2C8)⋊25C22, C2.45(C22×C5⋊D4), (C5×C4○D4).29C22, (C2×C4).103(C22×D5), SmallGroup(320,1508)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 790 in 258 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8 [×4], C2×C4 [×3], C2×C4 [×9], D4 [×6], D4 [×10], Q8 [×2], Q8 [×6], C23 [×3], D5, C10, C10 [×6], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×4], C4○D4, C4○D4 [×3], C4○D4 [×7], Dic5 [×3], C20, C20 [×3], C20, D10, C2×C10 [×3], C2×C10 [×6], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C5⋊2C8, C5⋊2C8 [×3], Dic10 [×3], Dic10 [×3], C4×D5 [×3], D20, C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×3], C5×D4 [×6], C5×D4 [×6], C5×Q8 [×2], C22×C10 [×3], D4○SD16, C2×C5⋊2C8 [×3], C4.Dic5 [×3], D4⋊D5 [×3], D4.D5 [×9], Q8⋊D5, C5⋊Q16 [×3], C2×Dic10 [×3], C4○D20 [×3], D4⋊2D5 [×3], Q8×D5, D4×C10 [×3], D4×C10 [×3], C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4, D4.D10 [×3], C2×D4.D5 [×3], Q8.Dic5, D4.8D10 [×3], D4.9D10 [×3], D4.10D10, C5×2+ (1+4), D20.33C23
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○SD16, C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4, D20.33C23
Generators and relations
G = < a,b,c,d,e | a20=b2=1, c2=d2=e2=a10, bab=a-1, ac=ca, ad=da, eae-1=a11, bc=cb, bd=db, ebe-1=a15b, dcd-1=a10c, ce=ec, 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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 46 51 56)(42 47 52 57)(43 48 53 58)(44 49 54 59)(45 50 55 60)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(1 35 11 25)(2 36 12 26)(3 37 13 27)(4 38 14 28)(5 39 15 29)(6 40 16 30)(7 21 17 31)(8 22 18 32)(9 23 19 33)(10 24 20 34)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(1 25 11 35)(2 36 12 26)(3 27 13 37)(4 38 14 28)(5 29 15 39)(6 40 16 30)(7 31 17 21)(8 22 18 32)(9 33 19 23)(10 24 20 34)(41 76 51 66)(42 67 52 77)(43 78 53 68)(44 69 54 79)(45 80 55 70)(46 71 56 61)(47 62 57 72)(48 73 58 63)(49 64 59 74)(50 75 60 65)
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,35,11,25)(2,36,12,26)(3,37,13,27)(4,38,14,28)(5,39,15,29)(6,40,16,30)(7,21,17,31)(8,22,18,32)(9,23,19,33)(10,24,20,34)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,25,11,35)(2,36,12,26)(3,27,13,37)(4,38,14,28)(5,29,15,39)(6,40,16,30)(7,31,17,21)(8,22,18,32)(9,33,19,23)(10,24,20,34)(41,76,51,66)(42,67,52,77)(43,78,53,68)(44,69,54,79)(45,80,55,70)(46,71,56,61)(47,62,57,72)(48,73,58,63)(49,64,59,74)(50,75,60,65)>;
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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)(41,46,51,56)(42,47,52,57)(43,48,53,58)(44,49,54,59)(45,50,55,60)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,35,11,25)(2,36,12,26)(3,37,13,27)(4,38,14,28)(5,39,15,29)(6,40,16,30)(7,21,17,31)(8,22,18,32)(9,23,19,33)(10,24,20,34)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (1,25,11,35)(2,36,12,26)(3,27,13,37)(4,38,14,28)(5,29,15,39)(6,40,16,30)(7,31,17,21)(8,22,18,32)(9,33,19,23)(10,24,20,34)(41,76,51,66)(42,67,52,77)(43,78,53,68)(44,69,54,79)(45,80,55,70)(46,71,56,61)(47,62,57,72)(48,73,58,63)(49,64,59,74)(50,75,60,65) );
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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,64),(22,63),(23,62),(24,61),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65)], [(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,26,31,36),(22,27,32,37),(23,28,33,38),(24,29,34,39),(25,30,35,40),(41,46,51,56),(42,47,52,57),(43,48,53,58),(44,49,54,59),(45,50,55,60),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)], [(1,35,11,25),(2,36,12,26),(3,37,13,27),(4,38,14,28),(5,39,15,29),(6,40,16,30),(7,21,17,31),(8,22,18,32),(9,23,19,33),(10,24,20,34),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(1,25,11,35),(2,36,12,26),(3,27,13,37),(4,38,14,28),(5,29,15,39),(6,40,16,30),(7,31,17,21),(8,22,18,32),(9,33,19,23),(10,24,20,34),(41,76,51,66),(42,67,52,77),(43,78,53,68),(44,69,54,79),(45,80,55,70),(46,71,56,61),(47,62,57,72),(48,73,58,63),(49,64,59,74),(50,75,60,65)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 23 | 0 | 0 | 0 | 0 | 0 |
| 35 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 |
| 33 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 26 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 0 | 0 | 26 | 26 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 1 | 0 | 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 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [23,35,0,0,0,0,0,25,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[14,33,0,0,0,0,9,27,0,0,0,0,0,0,0,0,26,15,0,0,0,0,26,26,0,0,15,15,0,0,0,0,26,15,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,40,0,0,0,0,0,0,1,0,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | ··· | 10T | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 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○SD16 | D20.33C23 |
| kernel | D20.33C23 | D4.D10 | C2×D4.D5 | Q8.Dic5 | D4.8D10 | D4.9D10 | D4.10D10 | 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}._{33}C_2^3 % in TeX
G:=Group("D20.33C2^3"); // GroupNames label
G:=SmallGroup(320,1508);
// by ID
G=gap.SmallGroup(320,1508);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,136,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=1,c^2=d^2=e^2=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^11,b*c=c*b,b*d=d*b,e*b*e^-1=a^15*b,d*c*d^-1=a^10*c,c*e=e*c,d*e=e*d>;
// generators/relations