direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C2×C20, C102.27C22, C20⋊3(C2×C10), C10⋊2(C2×C20), (C10×C20)⋊8C2, (C2×C20)⋊5C10, (C5×C20)⋊8C22, C5⋊2(C22×C20), (C2×Dic5)⋊5C10, Dic5⋊3(C2×C10), D10.8(C2×C10), (C2×C10).46D10, C52⋊10(C22×C4), C22.9(D5×C10), (C10×Dic5)⋊11C2, (C5×C10).20C23, C10.2(C22×C10), (C22×D5).4C10, C10.41(C22×D5), (C5×Dic5)⋊10C22, (D5×C10).25C22, C2.1(D5×C2×C10), (C5×C10)⋊9(C2×C4), (D5×C2×C10).8C2, (C2×C10).11(C2×C10), SmallGroup(400,182)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C20 |
Generators and relations for D5×C2×C20
G = < a,b,c,d | a2=b20=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 292 in 124 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C5, C2×C4, C2×C4, C23, D5, C10, C10, C10, C22×C4, Dic5, C20, C20, D10, C2×C10, C2×C10, C52, C4×D5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×C10, C5×D5, C5×C10, C5×C10, C2×C4×D5, C22×C20, C5×Dic5, C5×C20, D5×C10, C102, D5×C20, C10×Dic5, C10×C20, D5×C2×C10, D5×C2×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, D5, C10, C22×C4, C20, D10, C2×C10, C4×D5, C2×C20, C22×D5, C22×C10, C5×D5, C2×C4×D5, C22×C20, D5×C10, D5×C20, D5×C2×C10, D5×C2×C20
►On 80 points
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 79)(42 80)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)(41 53 45 57 49)(42 54 46 58 50)(43 55 47 59 51)(44 56 48 60 52)(61 73 65 77 69)(62 74 66 78 70)(63 75 67 79 71)(64 76 68 80 72)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 61)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
G:=sub<Sym(80)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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), (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,73,65,77,69)(62,74,66,78,70)(63,75,67,79,71)(64,76,68,80,72), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,61)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,79)(42,80)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78), (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), (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36)(41,53,45,57,49)(42,54,46,58,50)(43,55,47,59,51)(44,56,48,60,52)(61,73,65,77,69)(62,74,66,78,70)(63,75,67,79,71)(64,76,68,80,72), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,61)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(41,79),(42,80),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78)], [(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)], [(1,9,17,5,13),(2,10,18,6,14),(3,11,19,7,15),(4,12,20,8,16),(21,29,37,25,33),(22,30,38,26,34),(23,31,39,27,35),(24,32,40,28,36),(41,53,45,57,49),(42,54,46,58,50),(43,55,47,59,51),(44,56,48,60,52),(61,73,65,77,69),(62,74,66,78,70),(63,75,67,79,71),(64,76,68,80,72)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,61),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10L | 10M | ··· | 10AP | 10AQ | ··· | 10BF | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | D5 | D10 | D10 | C4×D5 | C5×D5 | D5×C10 | D5×C10 | D5×C20 |
| kernel | D5×C2×C20 | D5×C20 | C10×Dic5 | C10×C20 | D5×C2×C10 | D5×C10 | C2×C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 16 | 4 | 4 | 4 | 32 | 2 | 4 | 2 | 8 | 8 | 16 | 8 | 32 |
Matrix representation of D5×C2×C20 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 21 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 16 |
| 40 | 0 | 0 |
| 0 | 0 | 25 |
| 0 | 23 | 0 |
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[21,0,0,0,4,0,0,0,4],[1,0,0,0,18,0,0,0,16],[40,0,0,0,0,23,0,25,0] >;
D5×C2×C20 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_{20} % in TeX
G:=Group("D5xC2xC20"); // GroupNames label
G:=SmallGroup(400,182);
// by ID
G=gap.SmallGroup(400,182);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-5,194,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations