metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.39D4, M4(2)⋊6D10, Dic10.39D4, (C2×Q8)⋊4D10, C4○D4.7D10, (C5×D4).14D4, C4.106(D4×D5), (C5×Q8).14D4, C8.C22⋊1D5, C20.198(C2×D4), C5⋊4(D4.9D4), D20⋊7C4⋊11C2, (Q8×C10)⋊4C22, (C22×D5).6D4, C22.37(D4×D5), C10.65C22≀C2, D4⋊2Dic5⋊8C2, D4⋊8D10.2C2, C20.C23⋊5C2, C20.23D4⋊7C2, D4.11(C5⋊D4), (C2×C20).17C23, Q8.11(C5⋊D4), (C4×Dic5)⋊6C22, C4.Dic5⋊9C22, C20.46D4⋊11C2, C4○D20.25C22, C2.33(C23⋊D10), (C2×D20).134C22, (C5×M4(2))⋊16C22, C4.54(C2×C5⋊D4), (C2×C10).36(C2×D4), (C5×C8.C22)⋊5C2, (C2×C4).17(C22×D5), (C5×C4○D4).15C22, SmallGroup(320,829)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.39D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a13b, dbd=a18b, dcd=a10c-1 >
Subgroups: 750 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×8], C5, C8 [×2], C2×C4, C2×C4 [×6], D4, D4 [×9], Q8, Q8 [×3], C23 [×4], D5 [×3], C10, C10 [×2], C42, C22⋊C4 [×2], M4(2), M4(2), SD16 [×2], Q16 [×2], C2×D4 [×5], C2×Q8, C4○D4, C4○D4 [×3], Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, C2×C10, C4.D4, C4≀C2 [×2], C4.4D4, C8.C22, C8.C22, 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5 [×3], D20, D20 [×4], C2×Dic5, C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C22×D5 [×2], C22×D5 [×2], D4.9D4, C4.Dic5, C4×Dic5, D10⋊C4 [×2], Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5 [×3], Q8⋊2D5, Q8×C10, C5×C4○D4, C20.46D4, D20⋊7C4, D4⋊2Dic5, C20.C23, C20.23D4, C5×C8.C22, D4⋊8D10, D20.39D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4.9D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20.39D4
►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 32)(22 31)(23 30)(24 29)(25 28)(26 27)(33 40)(34 39)(35 38)(36 37)(41 51)(42 50)(43 49)(44 48)(45 47)(52 60)(53 59)(54 58)(55 57)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 67 22 54)(2 76 23 43)(3 65 24 52)(4 74 25 41)(5 63 26 50)(6 72 27 59)(7 61 28 48)(8 70 29 57)(9 79 30 46)(10 68 31 55)(11 77 32 44)(12 66 33 53)(13 75 34 42)(14 64 35 51)(15 73 36 60)(16 62 37 49)(17 71 38 58)(18 80 39 47)(19 69 40 56)(20 78 21 45)
(1 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(41 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 68)(62 67)(63 66)(64 65)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)
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)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,22,54)(2,76,23,43)(3,65,24,52)(4,74,25,41)(5,63,26,50)(6,72,27,59)(7,61,28,48)(8,70,29,57)(9,79,30,46)(10,68,31,55)(11,77,32,44)(12,66,33,53)(13,75,34,42)(14,64,35,51)(15,73,36,60)(16,62,37,49)(17,71,38,58)(18,80,39,47)(19,69,40,56)(20,78,21,45), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)>;
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)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(33,40)(34,39)(35,38)(36,37)(41,51)(42,50)(43,49)(44,48)(45,47)(52,60)(53,59)(54,58)(55,57)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,22,54)(2,76,23,43)(3,65,24,52)(4,74,25,41)(5,63,26,50)(6,72,27,59)(7,61,28,48)(8,70,29,57)(9,79,30,46)(10,68,31,55)(11,77,32,44)(12,66,33,53)(13,75,34,42)(14,64,35,51)(15,73,36,60)(16,62,37,49)(17,71,38,58)(18,80,39,47)(19,69,40,56)(20,78,21,45), (1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,68)(62,67)(63,66)(64,65)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75) );
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),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(33,40),(34,39),(35,38),(36,37),(41,51),(42,50),(43,49),(44,48),(45,47),(52,60),(53,59),(54,58),(55,57),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)], [(1,67,22,54),(2,76,23,43),(3,65,24,52),(4,74,25,41),(5,63,26,50),(6,72,27,59),(7,61,28,48),(8,70,29,57),(9,79,30,46),(10,68,31,55),(11,77,32,44),(12,66,33,53),(13,75,34,42),(14,64,35,51),(15,73,36,60),(16,62,37,49),(17,71,38,58),(18,80,39,47),(19,69,40,56),(20,78,21,45)], [(1,27),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(41,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,68),(62,67),(63,66),(64,65),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75)])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 20 | 20 | 20 | 2 | 2 | 4 | 8 | 20 | 20 | 20 | 2 | 2 | 8 | 40 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | D4.9D4 | D4×D5 | D4×D5 | D20.39D4 |
| kernel | D20.39D4 | C20.46D4 | D20⋊7C4 | D4⋊2Dic5 | C20.C23 | C20.23D4 | C5×C8.C22 | D4⋊8D10 | Dic10 | D20 | C5×D4 | C5×Q8 | C22×D5 | C8.C22 | M4(2) | C2×Q8 | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of D20.39D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 9 |
| 0 | 0 | 1 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 9 |
| 0 | 0 | 1 | 0 | 32 | 0 |
| 0 | 0 | 0 | 18 | 0 | 1 |
| 0 | 0 | 23 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 21 | 21 |
| 0 | 0 | 32 | 0 | 20 | 21 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,34,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,32,0,40,0,0,9,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,23,0,0,40,0,18,0,0,0,0,32,0,40,0,0,9,0,1,0],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,9,0,0,0,0,0,21,20,32,0,0,0,21,21,0,32],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D20.39D4 in GAP, Magma, Sage, TeX
D_{20}._{39}D_4 % in TeX
G:=Group("D20.39D4"); // GroupNames label
G:=SmallGroup(320,829);
// by ID
G=gap.SmallGroup(320,829);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,184,570,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^13*b,d*b*d=a^18*b,d*c*d=a^10*c^-1>;
// generators/relations