metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.29D4, C20.9C24, SD16⋊13D10, D40⋊21C22, C40.36C23, D20.5C23, Dic10.29D4, Dic20⋊18C22, Dic10.5C23, (C2×C8)⋊11D10, C4.76(D4×D5), C5⋊D4.9D4, D40⋊C2⋊5C2, (C2×C40)⋊6C22, D4⋊D5⋊2C22, (C2×Q8)⋊11D10, (D5×SD16)⋊5C2, (C2×SD16)⋊6D5, C20.84(C2×D4), C5⋊2(D4○SD16), (C8×D5)⋊9C22, Q8⋊D5⋊1C22, D4⋊6D10⋊6C2, D40⋊7C2⋊8C2, (Q8×D5)⋊1C22, C4.9(C23×D5), (C10×SD16)⋊2C2, D10.50(C2×D4), SD16⋊D5⋊5C2, C4○D20⋊4C22, C5⋊2C8.3C23, D4.D5⋊2C22, D4.7(C22×D5), (C5×D4).7C23, (C4×D5).5C23, (D4×D5).1C22, C5⋊Q16⋊1C22, C8.12(C22×D5), C22.21(D4×D5), SD16⋊3D5⋊5C2, (C2×D4).116D10, D20.3C4⋊5C2, D4.D10⋊8C2, (C5×Q8).3C23, Q8.3(C22×D5), C20.C23⋊7C2, C8⋊D5⋊10C22, C40⋊C2⋊19C22, Dic5.56(C2×D4), Q8⋊2D5⋊1C22, (Q8×C10)⋊19C22, (C2×C20).526C23, Q8.10D10⋊3C2, (C5×SD16)⋊14C22, D4⋊2D5.1C22, C10.110(C22×D4), C4.Dic5⋊29C22, (D4×C10).167C22, C2.83(C2×D4×D5), (C2×C10).399(C2×D4), (C2×C4).230(C22×D5), SmallGroup(320,1434)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1014 in 258 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×6], C22, C22 [×9], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×14], Q8 [×2], Q8 [×6], C23 [×3], D5 [×4], C10, C10 [×3], C2×C8, C2×C8 [×2], M4(2) [×3], D8 [×3], SD16 [×4], SD16 [×6], Q16 [×3], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×3], C4○D4 [×11], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×4], C2×C10, C2×C10 [×3], C8○D4, C2×SD16, C2×SD16 [×2], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C5⋊2C8 [×2], C40 [×2], Dic10 [×3], Dic10 [×2], C4×D5 [×2], C4×D5 [×6], D20 [×3], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×2], C5⋊D4 [×6], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×D5 [×2], C22×C10, D4○SD16, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], D40, Dic20, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C2×C40, C5×SD16 [×4], C4○D20 [×3], C4○D20 [×2], D4×D5 [×2], D4×D5, D4⋊2D5 [×2], D4⋊2D5, Q8×D5 [×2], Q8×D5, Q8⋊2D5 [×2], Q8⋊2D5, C2×C5⋊D4 [×2], D4×C10, Q8×C10, D20.3C4, D40⋊7C2, D5×SD16 [×2], D40⋊C2 [×2], SD16⋊D5 [×2], SD16⋊3D5 [×2], D4.D10, C20.C23, C10×SD16, D4⋊6D10, Q8.10D10, D20.29D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], D4○SD16, D4×D5 [×2], C23×D5, C2×D4×D5, D20.29D4
Generators and relations
G = < a,b,c,d | a20=b2=d2=1, c4=a10, bab=a-1, ac=ca, dad=a11, bc=cb, dbd=a10b, dcd=c3 >
►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 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(61 71)(62 70)(63 69)(64 68)(65 67)(72 80)(73 79)(74 78)(75 77)
(1 74 33 48 11 64 23 58)(2 75 34 49 12 65 24 59)(3 76 35 50 13 66 25 60)(4 77 36 51 14 67 26 41)(5 78 37 52 15 68 27 42)(6 79 38 53 16 69 28 43)(7 80 39 54 17 70 29 44)(8 61 40 55 18 71 30 45)(9 62 21 56 19 72 31 46)(10 63 22 57 20 73 32 47)
(1 28)(2 39)(3 30)(4 21)(5 32)(6 23)(7 34)(8 25)(9 36)(10 27)(11 38)(12 29)(13 40)(14 31)(15 22)(16 33)(17 24)(18 35)(19 26)(20 37)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(61 76)(62 67)(63 78)(64 69)(65 80)(66 71)(68 73)(70 75)(72 77)(74 79)
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,74,33,48,11,64,23,58)(2,75,34,49,12,65,24,59)(3,76,35,50,13,66,25,60)(4,77,36,51,14,67,26,41)(5,78,37,52,15,68,27,42)(6,79,38,53,16,69,28,43)(7,80,39,54,17,70,29,44)(8,61,40,55,18,71,30,45)(9,62,21,56,19,72,31,46)(10,63,22,57,20,73,32,47), (1,28)(2,39)(3,30)(4,21)(5,32)(6,23)(7,34)(8,25)(9,36)(10,27)(11,38)(12,29)(13,40)(14,31)(15,22)(16,33)(17,24)(18,35)(19,26)(20,37)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79)>;
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), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(61,71)(62,70)(63,69)(64,68)(65,67)(72,80)(73,79)(74,78)(75,77), (1,74,33,48,11,64,23,58)(2,75,34,49,12,65,24,59)(3,76,35,50,13,66,25,60)(4,77,36,51,14,67,26,41)(5,78,37,52,15,68,27,42)(6,79,38,53,16,69,28,43)(7,80,39,54,17,70,29,44)(8,61,40,55,18,71,30,45)(9,62,21,56,19,72,31,46)(10,63,22,57,20,73,32,47), (1,28)(2,39)(3,30)(4,21)(5,32)(6,23)(7,34)(8,25)(9,36)(10,27)(11,38)(12,29)(13,40)(14,31)(15,22)(16,33)(17,24)(18,35)(19,26)(20,37)(41,46)(42,57)(43,48)(44,59)(45,50)(47,52)(49,54)(51,56)(53,58)(55,60)(61,76)(62,67)(63,78)(64,69)(65,80)(66,71)(68,73)(70,75)(72,77)(74,79) );
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)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(61,71),(62,70),(63,69),(64,68),(65,67),(72,80),(73,79),(74,78),(75,77)], [(1,74,33,48,11,64,23,58),(2,75,34,49,12,65,24,59),(3,76,35,50,13,66,25,60),(4,77,36,51,14,67,26,41),(5,78,37,52,15,68,27,42),(6,79,38,53,16,69,28,43),(7,80,39,54,17,70,29,44),(8,61,40,55,18,71,30,45),(9,62,21,56,19,72,31,46),(10,63,22,57,20,73,32,47)], [(1,28),(2,39),(3,30),(4,21),(5,32),(6,23),(7,34),(8,25),(9,36),(10,27),(11,38),(12,29),(13,40),(14,31),(15,22),(16,33),(17,24),(18,35),(19,26),(20,37),(41,46),(42,57),(43,48),(44,59),(45,50),(47,52),(49,54),(51,56),(53,58),(55,60),(61,76),(62,67),(63,78),(64,69),(65,80),(66,71),(68,73),(70,75),(72,77),(74,79)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 0 | 0 | 1 | 24 |
| 0 | 0 | 17 | 38 |
| 40 | 17 | 0 | 0 |
| 24 | 3 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 26 | 0 | 15 | 0 |
| 0 | 26 | 0 | 15 |
| 26 | 0 | 26 | 0 |
| 0 | 26 | 0 | 26 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 24 | 1 |
| 0 | 0 | 40 | 17 |
G:=sub<GL(4,GF(41))| [0,0,40,24,0,0,17,3,1,17,0,0,24,38,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[26,0,26,0,0,26,0,26,15,0,26,0,0,15,0,26],[17,1,0,0,40,24,0,0,0,0,24,40,0,0,1,17] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D4○SD16 | D4×D5 | D4×D5 | D20.29D4 |
| kernel | D20.29D4 | D20.3C4 | D40⋊7C2 | D5×SD16 | D40⋊C2 | SD16⋊D5 | SD16⋊3D5 | D4.D10 | C20.C23 | C10×SD16 | D4⋊6D10 | Q8.10D10 | Dic10 | D20 | C5⋊D4 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_{20}._{29}D_4 % in TeX
G:=Group("D20.29D4"); // GroupNames label
G:=SmallGroup(320,1434);
// by ID
G=gap.SmallGroup(320,1434);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,185,136,438,235,102,12550]);
// 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,d*a*d=a^11,b*c=c*b,d*b*d=a^10*b,d*c*d=c^3>;
// generators/relations