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, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, C8.C22, 2+ 1+4, C5⋊2C8, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×D5, D4.9D4, C4.Dic5, C4×Dic5, D10⋊C4, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, 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, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, D4.9D4, D4×D5, 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 22)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 67)(62 66)(63 65)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)
(1 67 37 52)(2 76 38 41)(3 65 39 50)(4 74 40 59)(5 63 21 48)(6 72 22 57)(7 61 23 46)(8 70 24 55)(9 79 25 44)(10 68 26 53)(11 77 27 42)(12 66 28 51)(13 75 29 60)(14 64 30 49)(15 73 31 58)(16 62 32 47)(17 71 33 56)(18 80 34 45)(19 69 35 54)(20 78 36 43)
(1 22)(2 21)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(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)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,37,52)(2,76,38,41)(3,65,39,50)(4,74,40,59)(5,63,21,48)(6,72,22,57)(7,61,23,46)(8,70,24,55)(9,79,25,44)(10,68,26,53)(11,77,27,42)(12,66,28,51)(13,75,29,60)(14,64,30,49)(15,73,31,58)(16,62,32,47)(17,71,33,56)(18,80,34,45)(19,69,35,54)(20,78,36,43), (1,22)(2,21)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(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)(21,22)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,67)(62,66)(63,65)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75), (1,67,37,52)(2,76,38,41)(3,65,39,50)(4,74,40,59)(5,63,21,48)(6,72,22,57)(7,61,23,46)(8,70,24,55)(9,79,25,44)(10,68,26,53)(11,77,27,42)(12,66,28,51)(13,75,29,60)(14,64,30,49)(15,73,31,58)(16,62,32,47)(17,71,33,56)(18,80,34,45)(19,69,35,54)(20,78,36,43), (1,22)(2,21)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(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),(21,22),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,67),(62,66),(63,65),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75)], [(1,67,37,52),(2,76,38,41),(3,65,39,50),(4,74,40,59),(5,63,21,48),(6,72,22,57),(7,61,23,46),(8,70,24,55),(9,79,25,44),(10,68,26,53),(11,77,27,42),(12,66,28,51),(13,75,29,60),(14,64,30,49),(15,73,31,58),(16,62,32,47),(17,71,33,56),(18,80,34,45),(19,69,35,54),(20,78,36,43)], [(1,22),(2,21),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(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