metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.39C24, D20.34C23, 2- (1+4)⋊3D5, Dic10.34C23, C4○D4⋊6D10, (C2×Q8)⋊13D10, (C5×D4).38D4, C5⋊7(D4○SD16), (C5×Q8).38D4, D4⋊D5⋊22C22, C20.271(C2×D4), Q8⋊D5⋊20C22, D4⋊D10⋊12C2, D4⋊8D10⋊10C2, C4.39(C23×D5), D4.20(C5⋊D4), Q8.Dic5⋊12C2, C5⋊2C8.18C23, D4.D5⋊22C22, Q8.20(C5⋊D4), (Q8×C10)⋊23C22, (C5×D4).27C23, C5⋊Q16⋊19C22, D4.27(C22×D5), D4.8D10⋊10C2, (C5×Q8).27C23, Q8.27(C22×D5), C20.C23⋊11C2, (C2×C20).120C23, C4○D20.33C22, C10.173(C22×D4), C4.Dic5⋊18C22, (C5×2- (1+4))⋊2C2, (C2×D20).192C22, (C2×Q8⋊D5)⋊32C2, C4.77(C2×C5⋊D4), (C2×C10).87(C2×D4), (C5×C4○D4)⋊9C22, C22.8(C2×C5⋊D4), (C2×C5⋊2C8)⋊26C22, C2.46(C22×C5⋊D4), (C2×C4).104(C22×D5), SmallGroup(320,1509)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 918 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, D4 [×3], D4 [×12], Q8, Q8 [×3], Q8 [×4], C23 [×3], D5 [×3], C10, C10 [×4], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16 [×10], Q16 [×3], C2×D4 [×6], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×3], C4○D4 [×7], Dic5, C20, C20 [×3], C20 [×3], D10 [×6], C2×C10 [×3], C2×C10, 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, C4×D5 [×3], D20 [×3], D20 [×3], C5⋊D4 [×3], C2×C20 [×3], C2×C20 [×6], C5×D4, C5×D4 [×3], C5×D4 [×3], C5×Q8, C5×Q8 [×3], C5×Q8 [×3], C22×D5 [×3], D4○SD16, C2×C5⋊2C8 [×3], C4.Dic5 [×3], D4⋊D5 [×3], D4.D5, Q8⋊D5 [×9], C5⋊Q16 [×3], C2×D20 [×3], C4○D20 [×3], D4×D5 [×3], Q8⋊2D5, Q8×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×3], C5×C4○D4 [×3], C2×Q8⋊D5 [×3], C20.C23 [×3], Q8.Dic5, D4⋊D10 [×3], D4.8D10 [×3], D4⋊8D10, C5×2- (1+4), D20.34C23
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, D20.34C23
Generators and relations
G = < a,b,c,d,e | a20=b2=c2=d2=1, e2=a10, bab=dad=a-1, ac=ca, eae-1=a11, cbc=a10b, dbd=a18b, ebe-1=a15b, cd=dc, ce=ec, ede-1=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 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 61)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)
(1 37 11 27)(2 28 12 38)(3 39 13 29)(4 30 14 40)(5 21 15 31)(6 32 16 22)(7 23 17 33)(8 34 18 24)(9 25 19 35)(10 36 20 26)(41 61 51 71)(42 72 52 62)(43 63 53 73)(44 74 54 64)(45 65 55 75)(46 76 56 66)(47 67 57 77)(48 78 58 68)(49 69 59 79)(50 80 60 70)
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)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,61,51,71)(42,72,52,62)(43,63,53,73)(44,74,54,64)(45,65,55,75)(46,76,56,66)(47,67,57,77)(48,78,58,68)(49,69,59,79)(50,80,60,70)>;
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)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,41)(18,42)(19,43)(20,44)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62), (1,37,11,27)(2,28,12,38)(3,39,13,29)(4,30,14,40)(5,21,15,31)(6,32,16,22)(7,23,17,33)(8,34,18,24)(9,25,19,35)(10,36,20,26)(41,61,51,71)(42,72,52,62)(43,63,53,73)(44,74,54,64)(45,65,55,75)(46,76,56,66)(47,67,57,77)(48,78,58,68)(49,69,59,79)(50,80,60,70) );
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),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,41),(18,42),(19,43),(20,44),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,61),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62)], [(1,37,11,27),(2,28,12,38),(3,39,13,29),(4,30,14,40),(5,21,15,31),(6,32,16,22),(7,23,17,33),(8,34,18,24),(9,25,19,35),(10,36,20,26),(41,61,51,71),(42,72,52,62),(43,63,53,73),(44,74,54,64),(45,65,55,75),(46,76,56,66),(47,67,57,77),(48,78,58,68),(49,69,59,79),(50,80,60,70)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 5 | 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 | 16 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 35 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 34 | 11 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,5,0,0,0,0,16,40,0,0,0,0,0,0,40,36,0,0,0,0,25,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,16,40,0,0,0,0,0,0,1,5,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,7,0,0,0,0,35,0,0,0,0,34,0,0,0,0,6,0,0,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,0,0,0,0,35,11,0,0,30,34,0,0,0,0,35,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;
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 | ··· | 10L | 20A | ··· | 20T |
| 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 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 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○SD16 | D20.34C23 |
| kernel | D20.34C23 | C2×Q8⋊D5 | C20.C23 | Q8.Dic5 | D4⋊D10 | D4.8D10 | D4⋊8D10 | C5×2- (1+4) | C5×D4 | C5×Q8 | 2- (1+4) | C2×Q8 | 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}._{34}C_2^3 % in TeX
G:=Group("D20.34C2^3"); // GroupNames label
G:=SmallGroup(320,1509);
// by ID
G=gap.SmallGroup(320,1509);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,136,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^20=b^2=c^2=d^2=1,e^2=a^10,b*a*b=d*a*d=a^-1,a*c=c*a,e*a*e^-1=a^11,c*b*c=a^10*b,d*b*d=a^18*b,e*b*e^-1=a^15*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^5*d>;
// generators/relations