direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4×D5, C20⋊C23, C23⋊3D10, D20⋊7C22, D10⋊2C23, C10.5C24, Dic5⋊1C23, (C2×C4)⋊6D10, C10⋊2(C2×D4), (C2×C10)⋊C23, C5⋊2(C22×D4), (D4×C10)⋊5C2, C4⋊1(C22×D5), (C2×D20)⋊11C2, (C2×C20)⋊2C22, (C23×D5)⋊4C2, (C4×D5)⋊3C22, (C5×D4)⋊5C22, C5⋊D4⋊1C22, C2.6(C23×D5), C22⋊1(C22×D5), (C22×C10)⋊4C22, (C2×Dic5)⋊8C22, (C22×D5)⋊6C22, (C2×C4×D5)⋊3C2, (C2×C5⋊D4)⋊9C2, SmallGroup(160,217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4×D5
G = < a,b,c,d,e | a2=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 808 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, D4, C23, C23, D5, D5, C10, C10, C10, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C2×D4×D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, D4×D5, C23×D5, C2×D4×D5
►On 40 points
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 24 9 29)(2 25 10 30)(3 21 6 26)(4 22 7 27)(5 23 8 28)(11 31 16 36)(12 32 17 37)(13 33 18 38)(14 34 19 39)(15 35 20 40)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 18)(2 17)(3 16)(4 20)(5 19)(6 11)(7 15)(8 14)(9 13)(10 12)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)
G:=sub<Sym(40)| (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,24,9,29)(2,25,10,30)(3,21,6,26)(4,22,7,27)(5,23,8,28)(11,31,16,36)(12,32,17,37)(13,33,18,38)(14,34,19,39)(15,35,20,40), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)>;
G:=Group( (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,24,9,29)(2,25,10,30)(3,21,6,26)(4,22,7,27)(5,23,8,28)(11,31,16,36)(12,32,17,37)(13,33,18,38)(14,34,19,39)(15,35,20,40), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40), (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), (1,18)(2,17)(3,16)(4,20)(5,19)(6,11)(7,15)(8,14)(9,13)(10,12)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32) );
G=PermutationGroup([[(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,24,9,29),(2,25,10,30),(3,21,6,26),(4,22,7,27),(5,23,8,28),(11,31,16,36),(12,32,17,37),(13,33,18,38),(14,34,19,39),(15,35,20,40)], [(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40)], [(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)], [(1,18),(2,17),(3,16),(4,20),(5,19),(6,11),(7,15),(8,14),(9,13),(10,12),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32)]])
C2×D4×D5 is a maximal subgroup of
(D4×D5)⋊C4 D4⋊D20 D20.8D4 D20⋊D4 D10⋊6SD16 (D4×C10)⋊C4 (C2×D4)⋊7F5 D5⋊(C4.D4) (C2×F5)⋊D4 C2.(D4×F5) C42⋊11D10 D4⋊5D20 C24⋊3D10 C24⋊4D10 C10.372+ 1+4 C10.382+ 1+4 D20⋊19D4 C10.402+ 1+4 D20⋊20D4 C10.1202+ 1+4 C10.1212+ 1+4 C42⋊18D10 D20⋊10D4 C42⋊26D10 D20⋊11D4 C10.1452+ 1+4 D10.C24
C2×D4×D5 is a maximal quotient of
C24.27D10 C10.2- 1+4 C42⋊12D10 C42.228D10 D20⋊23D4 D20⋊24D4 Dic10⋊23D4 Dic10⋊24D4 C24.56D10 C24⋊3D10 C24⋊4D10 C24.33D10 C24.34D10 C20⋊(C4○D4) C10.682- 1+4 Dic10⋊19D4 Dic10⋊20D4 C10.372+ 1+4 C4⋊C4⋊21D10 C10.382+ 1+4 C10.392+ 1+4 D20⋊19D4 C10.402+ 1+4 C10.732- 1+4 D20⋊20D4 C4⋊C4⋊26D10 C10.162- 1+4 C10.172- 1+4 D20⋊21D4 D20⋊22D4 Dic10⋊21D4 Dic10⋊22D4 C10.792- 1+4 C10.1202+ 1+4 C10.1212+ 1+4 C10.822- 1+4 C4⋊C4⋊28D10 C42.233D10 C42⋊18D10 C42.141D10 D20⋊10D4 Dic10⋊10D4 C42⋊26D10 C42.238D10 D20⋊11D4 Dic10⋊11D4 C42.171D10 C42.240D10 D20⋊12D4 D20⋊8Q8 D8⋊13D10 D20.29D4 D20.30D4 Q16⋊D10 D8⋊15D10 D8⋊11D10 D20.47D4 SD16⋊D10 D8⋊5D10 D8⋊6D10 D40⋊C22 C40.C23 D20.44D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 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 | C2 | D4 | D5 | D10 | D10 | D10 | D4×D5 |
| kernel | C2×D4×D5 | C2×C4×D5 | C2×D20 | D4×D5 | C2×C5⋊D4 | D4×C10 | C23×D5 | D10 | C2×D4 | C2×C4 | D4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 4 | 2 | 2 | 8 | 4 | 4 |
Matrix representation of C2×D4×D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,1,0,0,0,0,1] >;
C2×D4×D5 in GAP, Magma, Sage, TeX
C_2\times D_4\times D_5
% in TeX
G:=Group("C2xD4xD5"); // GroupNames label
G:=SmallGroup(160,217);
// by ID
G=gap.SmallGroup(160,217);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,159,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations