metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊21D4, C10.1172+ (1+4), C4⋊C4⋊10D10, C22⋊Q8⋊7D5, C5⋊7(D4⋊5D4), (C2×Q8)⋊16D10, C4.111(D4×D5), C4⋊2D20⋊25C2, D10.43(C2×D4), C20.234(C2×D4), D20⋊8C4⋊25C2, C22⋊D20⋊16C2, (Q8×C10)⋊7C22, (C2×D20)⋊25C22, (C22×D20)⋊16C2, (C2×C20).54C23, C22⋊C4.57D10, C10.76(C22×D4), C20.23D4⋊12C2, (C2×C10).174C24, C22⋊2(Q8⋊2D5), (C4×Dic5)⋊28C22, (C22×C4).236D10, D10.13D4⋊17C2, C2.34(D4⋊8D10), D10⋊C4⋊20C22, C10.D4⋊53C22, (C23×D5).52C22, C23.189(C22×D5), C22.195(C23×D5), (C22×C20).254C22, (C22×C10).202C23, (C2×Dic5).243C23, (C22×D5).206C23, C23.D5.115C22, C2.49(C2×D4×D5), (C4×C5⋊D4)⋊22C2, (D5×C22⋊C4)⋊8C2, (C2×C4×D5)⋊18C22, (C2×C10)⋊7(C4○D4), (C5×C4⋊C4)⋊19C22, (C2×Q8⋊2D5)⋊7C2, (C5×C22⋊Q8)⋊10C2, C10.114(C2×C4○D4), C2.17(C2×Q8⋊2D5), (C2×C4).47(C22×D5), (C2×C5⋊D4).130C22, (C5×C22⋊C4).29C22, SmallGroup(320,1302)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1510 in 334 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×9], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×27], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×18], Q8 [×2], C23, C23 [×15], D5 [×7], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4, C22×C4, C22×C4 [×5], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×3], C20 [×2], C20 [×5], D10 [×4], D10 [×21], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×D5 [×8], D20 [×4], D20 [×12], C2×Dic5 [×3], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×4], C22×D5 [×10], C22×C10, D4⋊5D4, C4×Dic5, C10.D4, D10⋊C4, D10⋊C4 [×8], C23.D5, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C4×D5, C2×C4×D5 [×4], C2×D20 [×2], C2×D20 [×6], C2×D20 [×4], Q8⋊2D5 [×4], C2×C5⋊D4, C22×C20, Q8×C10, C23×D5 [×2], D5×C22⋊C4 [×2], C22⋊D20 [×2], D20⋊8C4, D10.13D4 [×2], C4⋊2D20, C4⋊2D20 [×2], C4×C5⋊D4, C20.23D4, C5×C22⋊Q8, C22×D20, C2×Q8⋊2D5, D20⋊21D4
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, D4×D5 [×2], Q8⋊2D5 [×2], C23×D5, C2×D4×D5, C2×Q8⋊2D5, D4⋊8D10, D20⋊21D4
Generators and relations
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, cac-1=dad=a9, cbc-1=dbd=a18b, 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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)
(1 63 60 33)(2 72 41 22)(3 61 42 31)(4 70 43 40)(5 79 44 29)(6 68 45 38)(7 77 46 27)(8 66 47 36)(9 75 48 25)(10 64 49 34)(11 73 50 23)(12 62 51 32)(13 71 52 21)(14 80 53 30)(15 69 54 39)(16 78 55 28)(17 67 56 37)(18 76 57 26)(19 65 58 35)(20 74 59 24)
(1 23)(2 32)(3 21)(4 30)(5 39)(6 28)(7 37)(8 26)(9 35)(10 24)(11 33)(12 22)(13 31)(14 40)(15 29)(16 38)(17 27)(18 36)(19 25)(20 34)(41 62)(42 71)(43 80)(44 69)(45 78)(46 67)(47 76)(48 65)(49 74)(50 63)(51 72)(52 61)(53 70)(54 79)(55 68)(56 77)(57 66)(58 75)(59 64)(60 73)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76), (1,63,60,33)(2,72,41,22)(3,61,42,31)(4,70,43,40)(5,79,44,29)(6,68,45,38)(7,77,46,27)(8,66,47,36)(9,75,48,25)(10,64,49,34)(11,73,50,23)(12,62,51,32)(13,71,52,21)(14,80,53,30)(15,69,54,39)(16,78,55,28)(17,67,56,37)(18,76,57,26)(19,65,58,35)(20,74,59,24), (1,23)(2,32)(3,21)(4,30)(5,39)(6,28)(7,37)(8,26)(9,35)(10,24)(11,33)(12,22)(13,31)(14,40)(15,29)(16,38)(17,27)(18,36)(19,25)(20,34)(41,62)(42,71)(43,80)(44,69)(45,78)(46,67)(47,76)(48,65)(49,74)(50,63)(51,72)(52,61)(53,70)(54,79)(55,68)(56,77)(57,66)(58,75)(59,64)(60,73)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,53)(42,52)(43,51)(44,50)(45,49)(46,48)(54,60)(55,59)(56,58)(61,69)(62,68)(63,67)(64,66)(70,80)(71,79)(72,78)(73,77)(74,76), (1,63,60,33)(2,72,41,22)(3,61,42,31)(4,70,43,40)(5,79,44,29)(6,68,45,38)(7,77,46,27)(8,66,47,36)(9,75,48,25)(10,64,49,34)(11,73,50,23)(12,62,51,32)(13,71,52,21)(14,80,53,30)(15,69,54,39)(16,78,55,28)(17,67,56,37)(18,76,57,26)(19,65,58,35)(20,74,59,24), (1,23)(2,32)(3,21)(4,30)(5,39)(6,28)(7,37)(8,26)(9,35)(10,24)(11,33)(12,22)(13,31)(14,40)(15,29)(16,38)(17,27)(18,36)(19,25)(20,34)(41,62)(42,71)(43,80)(44,69)(45,78)(46,67)(47,76)(48,65)(49,74)(50,63)(51,72)(52,61)(53,70)(54,79)(55,68)(56,77)(57,66)(58,75)(59,64)(60,73) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,53),(42,52),(43,51),(44,50),(45,49),(46,48),(54,60),(55,59),(56,58),(61,69),(62,68),(63,67),(64,66),(70,80),(71,79),(72,78),(73,77),(74,76)], [(1,63,60,33),(2,72,41,22),(3,61,42,31),(4,70,43,40),(5,79,44,29),(6,68,45,38),(7,77,46,27),(8,66,47,36),(9,75,48,25),(10,64,49,34),(11,73,50,23),(12,62,51,32),(13,71,52,21),(14,80,53,30),(15,69,54,39),(16,78,55,28),(17,67,56,37),(18,76,57,26),(19,65,58,35),(20,74,59,24)], [(1,23),(2,32),(3,21),(4,30),(5,39),(6,28),(7,37),(8,26),(9,35),(10,24),(11,33),(12,22),(13,31),(14,40),(15,29),(16,38),(17,27),(18,36),(19,25),(20,34),(41,62),(42,71),(43,80),(44,69),(45,78),(46,67),(47,76),(48,65),(49,74),(50,63),(51,72),(52,61),(53,70),(54,79),(55,68),(56,77),(57,66),(58,75),(59,64),(60,73)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 36 |
| 0 | 0 | 0 | 0 | 25 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 25 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 37 |
| 0 | 0 | 0 | 0 | 20 | 9 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 35 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 |
| 0 | 0 | 0 | 0 | 21 | 32 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,25,0,0,0,0,36,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,25,0,0,0,0,0,1],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,20,0,0,0,0,37,9],[40,35,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,21,0,0,0,0,4,32] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | Q8⋊2D5 | D4⋊8D10 |
| kernel | D20⋊21D4 | D5×C22⋊C4 | C22⋊D20 | D20⋊8C4 | D10.13D4 | C4⋊2D20 | C4×C5⋊D4 | C20.23D4 | C5×C22⋊Q8 | C22×D20 | C2×Q8⋊2D5 | D20 | C22⋊Q8 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 6 | 2 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{20}\rtimes_{21}D_4 % in TeX
G:=Group("D20:21D4"); // GroupNames label
G:=SmallGroup(320,1302);
// by ID
G=gap.SmallGroup(320,1302);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,192,12550]);
// 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=d*a*d=a^9,c*b*c^-1=d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations