metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.36D4, C4⋊C4⋊4D10, (C2×Q8)⋊1D10, C22⋊Q8⋊1D5, (C2×C10)⋊5SD16, C4.101(D4×D5), (C2×C20).77D4, C20.152(C2×D4), C5⋊3(C22⋊SD16), C22⋊2(Q8⋊D5), D20⋊6C4⋊37C2, (Q8×C10)⋊1C22, C10.47C22≀C2, C10.72(C2×SD16), (C22×C10).92D4, C20.55D4⋊13C2, (C2×C20).365C23, (C22×D20).14C2, (C22×C4).127D10, C23.62(C5⋊D4), C2.15(C23⋊D10), C2.15(D4⋊D10), C10.116(C8⋊C22), (C2×D20).250C22, (C22×C20).169C22, (C2×Q8⋊D5)⋊8C2, C2.9(C2×Q8⋊D5), (C5×C4⋊C4)⋊6C22, (C5×C22⋊Q8)⋊1C2, (C2×C5⋊2C8)⋊6C22, (C2×C10).496(C2×D4), (C2×C4).55(C5⋊D4), (C2×C4).465(C22×D5), C22.171(C2×C5⋊D4), SmallGroup(320,673)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.36D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd=a10c-1 >
Subgroups: 1006 in 188 conjugacy classes, 47 normal (27 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, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×7], C2×Q8, C24, C20 [×2], C20 [×3], D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C5⋊2C8 [×2], D20 [×4], D20 [×6], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×D5 [×10], C22×C10, C22⋊SD16, C2×C5⋊2C8 [×2], Q8⋊D5 [×4], C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×D20 [×2], C2×D20 [×5], C22×C20, Q8×C10, C23×D5, D20⋊6C4 [×2], C20.55D4, C2×Q8⋊D5 [×2], C5×C22⋊Q8, C22×D20, D20.36D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8⋊C22, C5⋊D4 [×2], C22×D5, C22⋊SD16, Q8⋊D5 [×2], D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C2×Q8⋊D5, D4⋊D10, D20.36D4
►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 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 58 28 62)(2 49 29 73)(3 60 30 64)(4 51 31 75)(5 42 32 66)(6 53 33 77)(7 44 34 68)(8 55 35 79)(9 46 36 70)(10 57 37 61)(11 48 38 72)(12 59 39 63)(13 50 40 74)(14 41 21 65)(15 52 22 76)(16 43 23 67)(17 54 24 78)(18 45 25 69)(19 56 26 80)(20 47 27 71)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(57 61)(58 62)(59 63)(60 64)
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,58,28,62)(2,49,29,73)(3,60,30,64)(4,51,31,75)(5,42,32,66)(6,53,33,77)(7,44,34,68)(8,55,35,79)(9,46,36,70)(10,57,37,61)(11,48,38,72)(12,59,39,63)(13,50,40,74)(14,41,21,65)(15,52,22,76)(16,43,23,67)(17,54,24,78)(18,45,25,69)(19,56,26,80)(20,47,27,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64)>;
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)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,58,28,62)(2,49,29,73)(3,60,30,64)(4,51,31,75)(5,42,32,66)(6,53,33,77)(7,44,34,68)(8,55,35,79)(9,46,36,70)(10,57,37,61)(11,48,38,72)(12,59,39,63)(13,50,40,74)(14,41,21,65)(15,52,22,76)(16,43,23,67)(17,54,24,78)(18,45,25,69)(19,56,26,80)(20,47,27,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80)(57,61)(58,62)(59,63)(60,64) );
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),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)], [(1,58,28,62),(2,49,29,73),(3,60,30,64),(4,51,31,75),(5,42,32,66),(6,53,33,77),(7,44,34,68),(8,55,35,79),(9,46,36,70),(10,57,37,61),(11,48,38,72),(12,59,39,63),(13,50,40,74),(14,41,21,65),(15,52,22,76),(16,43,23,67),(17,54,24,78),(18,45,25,69),(19,56,26,80),(20,47,27,71)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80),(57,61),(58,62),(59,63),(60,64)])
47 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 | ··· | 20P |
| 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 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4×D5 | Q8⋊D5 | D4⋊D10 |
| kernel | D20.36D4 | D20⋊6C4 | C20.55D4 | C2×Q8⋊D5 | C5×C22⋊Q8 | C22×D20 | D20 | C2×C20 | C22×C10 | C22⋊Q8 | C2×C10 | C4⋊C4 | C22×C4 | C2×Q8 | C2×C4 | C23 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 4 | 4 | 4 |
Matrix representation of D20.36D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 33 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 21 | 0 | 0 | 0 | 0 |
| 39 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 26 | 0 |
| 1 | 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 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,39,1],[0,39,0,0,0,0,21,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
D20.36D4 in GAP, Magma, Sage, TeX
D_{20}._{36}D_4 % in TeX
G:=Group("D20.36D4"); // GroupNames label
G:=SmallGroup(320,673);
// by ID
G=gap.SmallGroup(320,673);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=a^10*c^-1>;
// generators/relations