metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D20, Q8.12D20, D40:12C22, C40.11C23, C20.62C24, M4(2):21D10, D20.25C23, Dic20:21C22, Dic10.25C23, C8oD4:4D5, (C2xC8):7D10, C5:1(D4oD8), (C2xD40):15C2, (C5xD4).24D4, C20.74(C2xD4), C4.28(C2xD20), (C5xQ8).24D4, D4:8D10:4C2, C8:D10:12C2, (C2xC40):10C22, C4oD4.39D10, C4oD20:2C22, D40:7C2:12C2, C22.4(C2xD20), C8.53(C22xD5), C4.59(C23xD5), (C2xD20):31C22, C40:C2:12C22, C10.29(C22xD4), C2.31(C22xD20), (C2xC20).516C23, (C5xM4(2)):23C22, (C5xC8oD4):4C2, (C2xC10).9(C2xD4), (C5xC4oD4).46C22, (C2xC4).227(C22xD5), SmallGroup(320,1424)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.12D20
G = < a,b,c,d | a4=b2=d2=1, c20=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c19 >
Subgroups: 1286 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, D5, C10, C10, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C4oD4, C4oD4, Dic5, C20, C20, D10, C2xC10, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, C40, C40, Dic10, C4xD5, D20, D20, C5:D4, C2xC20, C5xD4, C5xQ8, C22xD5, D4oD8, C40:C2, D40, Dic20, C2xC40, C5xM4(2), C2xD20, C4oD20, D4xD5, Q8:2D5, C5xC4oD4, C2xD40, D40:7C2, C8:D10, C5xC8oD4, D4:8D10, D4.12D20
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, D20, C22xD5, D4oD8, C2xD20, C23xD5, C22xD20, D4.12D20
(1 41 21 61)(2 42 22 62)(3 43 23 63)(4 44 24 64)(5 45 25 65)(6 46 26 66)(7 47 27 67)(8 48 28 68)(9 49 29 69)(10 50 30 70)(11 51 31 71)(12 52 32 72)(13 53 33 73)(14 54 34 74)(15 55 35 75)(16 56 36 76)(17 57 37 77)(18 58 38 78)(19 59 39 79)(20 60 40 80)
(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 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
G:=sub<Sym(80)| (1,41,21,61)(2,42,22,62)(3,43,23,63)(4,44,24,64)(5,45,25,65)(6,46,26,66)(7,47,27,67)(8,48,28,68)(9,49,29,69)(10,50,30,70)(11,51,31,71)(12,52,32,72)(13,53,33,73)(14,54,34,74)(15,55,35,75)(16,56,36,76)(17,57,37,77)(18,58,38,78)(19,59,39,79)(20,60,40,80), (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)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)>;
G:=Group( (1,41,21,61)(2,42,22,62)(3,43,23,63)(4,44,24,64)(5,45,25,65)(6,46,26,66)(7,47,27,67)(8,48,28,68)(9,49,29,69)(10,50,30,70)(11,51,31,71)(12,52,32,72)(13,53,33,73)(14,54,34,74)(15,55,35,75)(16,56,36,76)(17,57,37,77)(18,58,38,78)(19,59,39,79)(20,60,40,80), (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)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61) );
G=PermutationGroup([[(1,41,21,61),(2,42,22,62),(3,43,23,63),(4,44,24,64),(5,45,25,65),(6,46,26,66),(7,47,27,67),(8,48,28,68),(9,49,29,69),(10,50,30,70),(11,51,31,71),(12,52,32,72),(13,53,33,73),(14,54,34,74),(15,55,35,75),(16,56,36,76),(17,57,37,77),(18,58,38,78),(19,59,39,79),(20,60,40,80)], [(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),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61)]])
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 | D4oD8 | D4.12D20 |
kernel | D4.12D20 | C2xD40 | D40:7C2 | C8:D10 | C5xC8oD4 | D4:8D10 | C5xD4 | C5xQ8 | C8oD4 | C2xC8 | M4(2) | C4oD4 | D4 | Q8 | C5 | C1 |
# reps | 1 | 3 | 3 | 6 | 1 | 2 | 3 | 1 | 2 | 6 | 6 | 2 | 12 | 4 | 2 | 8 |
Matrix representation of D4.12D20 ►in 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] >;
D4.12D20 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