metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.31D4, C23.36D20, (C2×C8)⋊15D10, C22⋊C8⋊8D5, (C2×C10)⋊1SD16, C4.119(D4×D5), (C2×C20).42D4, (C2×C4).31D20, D20⋊5C4⋊8C2, (C2×C40)⋊14C22, C10.7C22≀C2, C20.331(C2×D4), C5⋊1(C22⋊SD16), C4⋊Dic5⋊1C22, C10.7(C2×SD16), C20.48D4⋊1C2, C10.8(C8⋊C22), C22⋊3(C40⋊C2), (C22×D20).3C2, (C22×C4).81D10, (C22×C10).51D4, C2.11(C8⋊D10), (C2×C20).741C23, (C2×Dic10)⋊1C22, C22.104(C2×D20), C2.10(C22⋊D20), (C2×D20).197C22, (C22×C20).50C22, (C2×C40⋊C2)⋊9C2, (C5×C22⋊C8)⋊10C2, C2.10(C2×C40⋊C2), (C2×C10).124(C2×D4), (C2×C4).686(C22×D5), SmallGroup(320,358)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.31D4
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=a15c3 >
Subgroups: 1070 in 188 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×10], Q8 [×2], C23, C23 [×10], D5 [×4], C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×7], C2×Q8, C24, Dic5 [×2], C20 [×2], C20, D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C40 [×2], Dic10 [×2], D20 [×4], D20 [×6], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×10], C22×C10, C22⋊SD16, C40⋊C2 [×4], C10.D4, C4⋊Dic5, C23.D5, C2×C40 [×2], C2×Dic10, C2×D20 [×2], C2×D20 [×5], C22×C20, C23×D5, D20⋊5C4 [×2], C5×C22⋊C8, C2×C40⋊C2 [×2], C20.48D4, C22×D20, D20.31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, D20 [×2], C22×D5, C22⋊SD16, C40⋊C2 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×C40⋊C2, C8⋊D10, D20.31D4
►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 27)(22 26)(23 25)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 22 55 79 11 32 45 69)(2 23 56 80 12 33 46 70)(3 24 57 61 13 34 47 71)(4 25 58 62 14 35 48 72)(5 26 59 63 15 36 49 73)(6 27 60 64 16 37 50 74)(7 28 41 65 17 38 51 75)(8 29 42 66 18 39 52 76)(9 30 43 67 19 40 53 77)(10 31 44 68 20 21 54 78)
(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)
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)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,22,55,79,11,32,45,69)(2,23,56,80,12,33,46,70)(3,24,57,61,13,34,47,71)(4,25,58,62,14,35,48,72)(5,26,59,63,15,36,49,73)(6,27,60,64,16,37,50,74)(7,28,41,65,17,38,51,75)(8,29,42,66,18,39,52,76)(9,30,43,67,19,40,53,77)(10,31,44,68,20,21,54,78), (21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)>;
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)(21,27)(22,26)(23,25)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,22,55,79,11,32,45,69)(2,23,56,80,12,33,46,70)(3,24,57,61,13,34,47,71)(4,25,58,62,14,35,48,72)(5,26,59,63,15,36,49,73)(6,27,60,64,16,37,50,74)(7,28,41,65,17,38,51,75)(8,29,42,66,18,39,52,76)(9,30,43,67,19,40,53,77)(10,31,44,68,20,21,54,78), (21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72) );
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),(21,27),(22,26),(23,25),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)], [(1,22,55,79,11,32,45,69),(2,23,56,80,12,33,46,70),(3,24,57,61,13,34,47,71),(4,25,58,62,14,35,48,72),(5,26,59,63,15,36,49,73),(6,27,60,64,16,37,50,74),(7,28,41,65,17,38,51,75),(8,29,42,66,18,39,52,76),(9,30,43,67,19,40,53,77),(10,31,44,68,20,21,54,78)], [(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72)])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D20 | D20 | C40⋊C2 | C8⋊C22 | D4×D5 | C8⋊D10 |
| kernel | D20.31D4 | D20⋊5C4 | C5×C22⋊C8 | C2×C40⋊C2 | C20.48D4 | C22×D20 | D20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 | C10 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of D20.31D4 ►in GL6(𝔽41)
| 10 | 5 | 0 | 0 | 0 | 0 |
| 29 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 13 | 31 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 29 | 7 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 36 |
| 0 | 0 | 0 | 0 | 31 | 19 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 40 |
G:=sub<GL(6,GF(41))| [10,29,0,0,0,0,5,31,0,0,0,0,0,0,35,5,0,0,0,0,1,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[10,13,0,0,0,0,5,31,0,0,0,0,0,0,40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,24,0,0,0,0,0,1],[29,16,0,0,0,0,7,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,22,31,0,0,0,0,36,19],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40] >;
D20.31D4 in GAP, Magma, Sage, TeX
D_{20}._{31}D_4 % in TeX
G:=Group("D20.31D4"); // GroupNames label
G:=SmallGroup(320,358);
// by ID
G=gap.SmallGroup(320,358);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=d^2=1,c^4=a^10,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=a^15*c^3>;
// generators/relations