direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4≀C2, C42⋊31D10, M4(2)⋊15D10, (D4×D5)⋊3C4, D4⋊7(C4×D5), Q8⋊7(C4×D5), (Q8×D5)⋊3C4, D4⋊2D5⋊3C4, D20⋊15(C2×C4), (C4×C20)⋊9C22, Q8⋊2D5⋊3C4, (C4×D5).71D4, C4.200(D4×D5), (D5×C42)⋊1C2, D20⋊7C4⋊5C2, D20⋊4C4⋊3C2, C4○D4.18D10, C20.359(C2×D4), (D5×M4(2))⋊8C2, C22.27(D4×D5), D4⋊2Dic5⋊1C2, Dic10⋊14(C2×C4), C20.52(C22×C4), C4○D20.9C22, C4.Dic5⋊2C22, (C2×C20).260C23, (C2×Dic5).215D4, (C4×Dic5)⋊61C22, (C22×D5).117D4, D10.57(C22⋊C4), (C5×M4(2))⋊13C22, Dic5.27(C22⋊C4), C5⋊4(C2×C4≀C2), (C5×C4≀C2)⋊5C2, C4.17(C2×C4×D5), (C5×D4)⋊15(C2×C4), (C5×Q8)⋊14(C2×C4), (D5×C4○D4).1C2, (C4×D5).52(C2×C4), (C2×C10).24(C2×D4), C2.25(D5×C22⋊C4), C10.65(C2×C22⋊C4), (C5×C4○D4).1C22, (C2×C4×D5).300C22, (C2×C4).367(C22×D5), SmallGroup(320,447)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4≀C2
G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >
Subgroups: 686 in 170 conjugacy classes, 53 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C4≀C2, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C2×C4≀C2, C8×D5, C8⋊D5, C4.Dic5, C4×Dic5, C4×Dic5, C4×C20, C5×M4(2), C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D20⋊4C4, D20⋊7C4, D4⋊2Dic5, C5×C4≀C2, D5×C42, D5×M4(2), D5×C4○D4, D5×C4≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4≀C2, C2×C22⋊C4, C4×D5, C22×D5, C2×C4≀C2, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×C4≀C2
►On 40 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)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
G:=sub<Sym(40)| (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)>;
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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40) );
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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 4P | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 20 | 1 | 1 | 2 | ··· | 2 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4≀C2 | C4×D5 | C4×D5 | D4×D5 | D4×D5 | D5×C4≀C2 |
| kernel | D5×C4≀C2 | D20⋊4C4 | D20⋊7C4 | D4⋊2Dic5 | C5×C4≀C2 | D5×C42 | D5×M4(2) | D5×C4○D4 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 | C4×D5 | C2×Dic5 | C22×D5 | C4≀C2 | C42 | M4(2) | C4○D4 | D5 | D4 | Q8 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 8 |
Matrix representation of D5×C4≀C2 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 1 |
| 0 | 0 | 33 | 7 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 33 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 32 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,33,0,0,1,7],[40,0,0,0,0,40,0,0,0,0,40,33,0,0,0,1],[32,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,32,0,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
D5×C4≀C2 in GAP, Magma, Sage, TeX
D_5\times C_4\wr C_2
% in TeX
G:=Group("D5xC4wrC2"); // GroupNames label
G:=SmallGroup(320,447);
// by ID
G=gap.SmallGroup(320,447);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,136,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
// generators/relations