metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊23D4, C42⋊14D10, C10.1032+ (1+4), C4⋊C4⋊47D10, (C4×D4)⋊12D5, (C4×D20)⋊28C2, (D4×C20)⋊14C2, C5⋊2(D4⋊5D4), C4.139(D4×D5), D10⋊D4⋊7C2, C20⋊7D4⋊18C2, (C4×C20)⋊19C22, C22⋊C4⋊46D10, D10.37(C2×D4), C20.345(C2×D4), (C22×D20)⋊9C2, (C22×C4)⋊12D10, C23⋊D10⋊20C2, (C2×D4).213D10, C4.D20⋊16C2, C22⋊3(C4○D20), (C2×C10).94C24, C4⋊Dic5⋊59C22, C10.49(C22×D4), D10.13D4⋊7C2, C20.48D4⋊10C2, (C2×C20).782C23, (C22×C20)⋊16C22, C10.D4⋊3C22, C2.15(D4⋊8D10), D10⋊C4⋊30C22, C23.94(C22×D5), (C2×Dic10)⋊53C22, (C2×D20).218C22, (D4×C10).305C22, (C2×Dic5).40C23, (C23×D5).39C22, C22.119(C23×D5), C23.D5.11C22, (C22×C10).164C23, (C22×D5).182C23, C2.22(C2×D4×D5), (C2×C4○D20)⋊7C2, (C2×C4×D5)⋊48C22, (C2×C10)⋊2(C4○D4), (C5×C4⋊C4)⋊59C22, (D5×C22⋊C4)⋊28C2, C10.41(C2×C4○D4), C2.45(C2×C4○D20), (C2×C5⋊D4)⋊3C22, (C5×C22⋊C4)⋊56C22, (C2×C4).158(C22×D5), SmallGroup(320,1222)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1462 in 334 conjugacy classes, 107 normal (51 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C5, C2×C4 [×5], C2×C4 [×14], D4 [×18], Q8 [×2], C23 [×2], C23 [×14], D5 [×6], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×12], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×4], C20 [×2], C20 [×4], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C4×D4, C4×D4, C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×6], D20 [×4], D20 [×6], C2×Dic5 [×4], C5⋊D4 [×6], C2×C20 [×5], C2×C20 [×4], C5×D4 [×2], C22×D5 [×4], C22×D5 [×10], C22×C10 [×2], D4⋊5D4, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×2], C2×D20 [×4], C4○D20 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], D4×C10, C23×D5 [×2], C4×D20, C4.D20, D5×C22⋊C4 [×2], D10⋊D4 [×2], D10.13D4 [×2], C20.48D4, C20⋊7D4, C23⋊D10 [×2], D4×C20, C22×D20, C2×C4○D20, D20⋊23D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊5D4, C4○D20 [×2], D4×D5 [×2], C23×D5, C2×C4○D20, C2×D4×D5, D4⋊8D10, D20⋊23D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a10b, bd=db, dcd=c-1 >
►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 42)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)
(1 77 32 57)(2 78 33 58)(3 79 34 59)(4 80 35 60)(5 61 36 41)(6 62 37 42)(7 63 38 43)(8 64 39 44)(9 65 40 45)(10 66 21 46)(11 67 22 47)(12 68 23 48)(13 69 24 49)(14 70 25 50)(15 71 26 51)(16 72 27 52)(17 73 28 53)(18 74 29 54)(19 75 30 55)(20 76 31 56)
(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
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)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72), (1,77,32,57)(2,78,33,58)(3,79,34,59)(4,80,35,60)(5,61,36,41)(6,62,37,42)(7,63,38,43)(8,64,39,44)(9,65,40,45)(10,66,21,46)(11,67,22,47)(12,68,23,48)(13,69,24,49)(14,70,25,50)(15,71,26,51)(16,72,27,52)(17,73,28,53)(18,74,29,54)(19,75,30,55)(20,76,31,56), (41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)>;
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)(41,42)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,62)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72), (1,77,32,57)(2,78,33,58)(3,79,34,59)(4,80,35,60)(5,61,36,41)(6,62,37,42)(7,63,38,43)(8,64,39,44)(9,65,40,45)(10,66,21,46)(11,67,22,47)(12,68,23,48)(13,69,24,49)(14,70,25,50)(15,71,26,51)(16,72,27,52)(17,73,28,53)(18,74,29,54)(19,75,30,55)(20,76,31,56), (41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80) );
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),(41,42),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,62),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72)], [(1,77,32,57),(2,78,33,58),(3,79,34,59),(4,80,35,60),(5,61,36,41),(6,62,37,42),(7,63,38,43),(8,64,39,44),(9,65,40,45),(10,66,21,46),(11,67,22,47),(12,68,23,48),(13,69,24,49),(14,70,25,50),(15,71,26,51),(16,72,27,52),(17,73,28,53),(18,74,29,54),(19,75,30,55),(20,76,31,56)], [(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 14 | 11 |
| 0 | 0 | 30 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 27 |
| 0 | 0 | 32 | 11 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,14,30,0,0,11,9],[1,0,0,0,0,1,0,0,0,0,30,32,0,0,27,11],[0,1,0,0,40,0,0,0,0,0,17,40,0,0,1,24],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4×D5 | D4⋊8D10 |
| kernel | D20⋊23D4 | C4×D20 | C4.D20 | D5×C22⋊C4 | D10⋊D4 | D10.13D4 | C20.48D4 | C20⋊7D4 | C23⋊D10 | D4×C20 | C22×D20 | C2×C4○D20 | D20 | C4×D4 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{23}D_4 % in TeX
G:=Group("D20:23D4"); // GroupNames label
G:=SmallGroup(320,1222);
// by ID
G=gap.SmallGroup(320,1222);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^10*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations