direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C22⋊C4, D10.21D4, C23.13D10, (C2×C4)⋊5D10, C2.1(D4×D5), D10⋊6(C2×C4), C22⋊3(C4×D5), (C2×C20)⋊6C22, C10.17(C2×D4), (C22×D5)⋊3C4, C23.D5⋊3C2, D10⋊C4⋊9C2, (C23×D5).1C2, (C2×C10).21C23, C10.19(C22×C4), (C2×Dic5)⋊5C22, C22.13(C22×D5), (C22×C10).10C22, (C22×D5).42C22, (C2×C4×D5)⋊8C2, C2.8(C2×C4×D5), C5⋊2(C2×C22⋊C4), (C2×C10)⋊4(C2×C4), (C5×C22⋊C4)⋊8C2, SmallGroup(160,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C22⋊C4
G = < a,b,c,d,e | a5=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 472 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C10, C22⋊C4, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, D5×C22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, C22×D5, C2×C4×D5, D4×D5, D5×C22⋊C4
►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)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)
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)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)>;
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)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30) );
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),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30)]])
D5×C22⋊C4 is a maximal subgroup of
(C22×F5)⋊C4 D10⋊(C4⋊C4) C10.(C4×D4) C24.24D10 C24.27D10 C42⋊7D10 C42⋊10D10 C4×D4×D5 C42⋊11D10 D20⋊23D4 C42⋊16D10 C24⋊3D10 C24.33D10 C10.402+ 1+4 D20⋊20D4 C10.422+ 1+4 D20⋊21D4 C10.512+ 1+4 C10.532+ 1+4 C10.1202+ 1+4 C10.1212+ 1+4 C10.612+ 1+4 C10.1222+ 1+4 C10.622+ 1+4 D20⋊10D4 C42⋊20D10 C42⋊21D10 C42⋊23D10 C42⋊24D10 D30.27D4 D30.45D4
D5×C22⋊C4 is a maximal quotient of
(C2×C20)⋊Q8 C22.58(D4×D5) (C2×C4)⋊9D20 D10⋊2C42 D10⋊2(C4⋊C4) D10⋊7M4(2) C22⋊C8⋊D5 C23⋊C4⋊5D5 M4(2).19D10 M4(2).21D10 (D4×D5)⋊C4 D4⋊(C4×D5) D4⋊2D5⋊C4 (Q8×D5)⋊C4 Q8⋊(C4×D5) Q8⋊2D5⋊C4 C42⋊D10 C24.44D10 C24.48D10 C24.12D10 C24.13D10 D30.27D4 D30.45D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 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 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D4×D5 |
| kernel | D5×C22⋊C4 | D10⋊C4 | C23.D5 | C5×C22⋊C4 | C2×C4×D5 | C23×D5 | C22×D5 | D10 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 2 | 4 | 2 | 8 | 4 |
Matrix representation of D5×C22⋊C4 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 32 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 39 |
| 0 | 0 | 40 | 9 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,32,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,32,40,0,0,39,9] >;
D5×C22⋊C4 in GAP, Magma, Sage, TeX
D_5\times C_2^2\rtimes C_4
% in TeX
G:=Group("D5xC2^2:C4"); // GroupNames label
G:=SmallGroup(160,101);
// by ID
G=gap.SmallGroup(160,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations