metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D20, Q8.12D20, D40⋊12C22, C20.62C24, C40.11C23, M4(2)⋊21D10, D20.25C23, Dic20⋊21C22, Dic10.25C23, C8○D4⋊4D5, (C2×C8)⋊7D10, C5⋊1(D4○D8), (C2×D40)⋊15C2, (C5×D4).24D4, C4.28(C2×D20), C20.74(C2×D4), (C5×Q8).24D4, D4⋊8D10⋊4C2, C8⋊D10⋊12C2, (C2×C40)⋊10C22, C4○D4.39D10, C4○D20⋊2C22, D40⋊7C2⋊12C2, C4.59(C23×D5), C22.4(C2×D20), C8.53(C22×D5), (C2×D20)⋊31C22, C40⋊C2⋊12C22, C2.31(C22×D20), C10.29(C22×D4), (C2×C20).516C23, (C5×M4(2))⋊23C22, (C5×C8○D4)⋊4C2, (C2×C10).9(C2×D4), (C5×C4○D4).46C22, (C2×C4).227(C22×D5), SmallGroup(320,1424)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1286 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], D5 [×6], C10, C10 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], Dic5 [×2], C20, C20 [×3], D10 [×12], C2×C10 [×3], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C40, C40 [×3], Dic10 [×2], C4×D5 [×6], D20 [×6], D20 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C22×D5 [×6], D4○D8, C40⋊C2 [×6], D40 [×9], Dic20, C2×C40 [×3], C5×M4(2) [×3], C2×D20 [×6], C4○D20 [×6], D4×D5 [×6], Q8⋊2D5 [×2], C5×C4○D4, C2×D40 [×3], D40⋊7C2 [×3], C8⋊D10 [×6], C5×C8○D4, D4⋊8D10 [×2], D4.12D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, D20 [×4], C22×D5 [×7], D4○D8, C2×D20 [×6], C23×D5, C22×D20, D4.12D20
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c19 >
►On 80 points
(1 48 21 68)(2 49 22 69)(3 50 23 70)(4 51 24 71)(5 52 25 72)(6 53 26 73)(7 54 27 74)(8 55 28 75)(9 56 29 76)(10 57 30 77)(11 58 31 78)(12 59 32 79)(13 60 33 80)(14 61 34 41)(15 62 35 42)(16 63 36 43)(17 64 37 44)(18 65 38 45)(19 66 39 46)(20 67 40 47)
(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)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 48)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)
G:=sub<Sym(80)| (1,48,21,68)(2,49,22,69)(3,50,23,70)(4,51,24,71)(5,52,25,72)(6,53,26,73)(7,54,27,74)(8,55,28,75)(9,56,29,76)(10,57,30,77)(11,58,31,78)(12,59,32,79)(13,60,33,80)(14,61,34,41)(15,62,35,42)(16,63,36,43)(17,64,37,44)(18,65,38,45)(19,66,39,46)(20,67,40,47), (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), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)>;
G:=Group( (1,48,21,68)(2,49,22,69)(3,50,23,70)(4,51,24,71)(5,52,25,72)(6,53,26,73)(7,54,27,74)(8,55,28,75)(9,56,29,76)(10,57,30,77)(11,58,31,78)(12,59,32,79)(13,60,33,80)(14,61,34,41)(15,62,35,42)(16,63,36,43)(17,64,37,44)(18,65,38,45)(19,66,39,46)(20,67,40,47), (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), (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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68) );
G=PermutationGroup([(1,48,21,68),(2,49,22,69),(3,50,23,70),(4,51,24,71),(5,52,25,72),(6,53,26,73),(7,54,27,74),(8,55,28,75),(9,56,29,76),(10,57,30,77),(11,58,31,78),(12,59,32,79),(13,60,33,80),(14,61,34,41),(15,62,35,42),(16,63,36,43),(17,64,37,44),(18,65,38,45),(19,66,39,46),(20,67,40,47)], [(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)], [(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)], [(1,40),(2,39),(3,38),(4,37),(5,36),(6,35),(7,34),(8,33),(9,32),(10,31),(11,30),(12,29),(13,28),(14,27),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(41,54),(42,53),(43,52),(44,51),(45,50),(46,49),(47,48),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 20 | 0 |
| 0 | 0 | 38 | 0 | 27 | 40 |
| 0 | 0 | 4 | 0 | 40 | 0 |
| 0 | 0 | 23 | 1 | 36 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 40 | 0 |
| 0 | 0 | 20 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 13 | 0 | 0 |
| 0 | 0 | 8 | 18 | 0 | 0 |
| 0 | 0 | 37 | 26 | 35 | 26 |
| 0 | 0 | 23 | 7 | 12 | 23 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 13 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 26 | 35 | 26 |
| 0 | 0 | 21 | 7 | 16 | 6 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,38,4,23,0,0,0,0,0,1,0,0,20,27,40,36,0,0,0,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,4,20,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,35,0,0,0,0,0,0,40,8,37,23,0,0,13,18,26,7,0,0,0,0,35,12,0,0,0,0,26,23],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,37,21,0,0,13,1,26,7,0,0,0,0,35,16,0,0,0,0,26,6] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D20 | D4○D8 | D4.12D20 |
| kernel | D4.12D20 | C2×D40 | D40⋊7C2 | C8⋊D10 | C5×C8○D4 | D4⋊8D10 | C5×D4 | C5×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
D_4._{12}D_{20} % in TeX
G:=Group("D4.12D20"); // GroupNames label
G:=SmallGroup(320,1424);
// by ID
G=gap.SmallGroup(320,1424);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^20=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^19>;
// generators/relations