metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2D10, C5⋊2C2≀C22, (C2×C20)⋊2D4, C22≀C2⋊1D5, C22⋊C4⋊3D10, (C2×Dic5)⋊2D4, (C22×D5)⋊2D4, (C22×C10)⋊3D4, D4⋊6D10⋊3C2, (C2×D4).33D10, C24⋊2D5⋊1C2, C23⋊1(C5⋊D4), C23⋊Dic5⋊5C2, C22.33(D4×D5), C10.43C22≀C2, (C23×C10)⋊7C22, C23.D5⋊4C22, (D4×C10).49C22, C23.1D10⋊5C2, C2.11(C23⋊D10), C23.74(C22×D5), (C22×C10).113C23, (C2×C4)⋊1(C5⋊D4), (C5×C22≀C2)⋊1C2, (C2×C10).30(C2×D4), (C2×C5⋊D4).5C22, C22.29(C2×C5⋊D4), (C5×C22⋊C4)⋊34C22, SmallGroup(320,659)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=abcd, bc=cb, ebe-1=fbf=bd=db, fcf=cd=dc, ce=ec, de=ed, df=fd, fef=e-1 >
Subgroups: 926 in 198 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C23⋊C4, C22≀C2, C22≀C2, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C2≀C22, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D4×C10, C23×C10, C23.1D10, C23⋊Dic5, C24⋊2D5, C5×C22≀C2, D4⋊6D10, C24⋊2D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C2≀C22, D4×D5, C2×C5⋊D4, C23⋊D10, C24⋊2D10
►On 40 points
(1 33)(2 27)(3 35)(4 29)(5 37)(6 21)(7 39)(8 23)(9 31)(10 25)(11 24)(12 32)(13 26)(14 34)(15 28)(16 36)(17 30)(18 38)(19 22)(20 40)
(1 26)(2 14)(3 28)(4 16)(5 30)(6 18)(7 22)(8 20)(9 24)(10 12)(11 31)(13 33)(15 35)(17 37)(19 39)(21 38)(23 40)(25 32)(27 34)(29 36)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 38)(22 39)(23 40)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 21)(19 22)(20 23)
(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 5)(2 4)(6 10)(7 9)(11 22)(12 21)(13 30)(14 29)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(31 39)(32 38)(33 37)(34 36)
G:=sub<Sym(40)| (1,33)(2,27)(3,35)(4,29)(5,37)(6,21)(7,39)(8,23)(9,31)(10,25)(11,24)(12,32)(13,26)(14,34)(15,28)(16,36)(17,30)(18,38)(19,22)(20,40), (1,26)(2,14)(3,28)(4,16)(5,30)(6,18)(7,22)(8,20)(9,24)(10,12)(11,31)(13,33)(15,35)(17,37)(19,39)(21,38)(23,40)(25,32)(27,34)(29,36), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(21,38)(22,39)(23,40)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,21)(19,22)(20,23), (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), (1,5)(2,4)(6,10)(7,9)(11,22)(12,21)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(31,39)(32,38)(33,37)(34,36)>;
G:=Group( (1,33)(2,27)(3,35)(4,29)(5,37)(6,21)(7,39)(8,23)(9,31)(10,25)(11,24)(12,32)(13,26)(14,34)(15,28)(16,36)(17,30)(18,38)(19,22)(20,40), (1,26)(2,14)(3,28)(4,16)(5,30)(6,18)(7,22)(8,20)(9,24)(10,12)(11,31)(13,33)(15,35)(17,37)(19,39)(21,38)(23,40)(25,32)(27,34)(29,36), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,11)(10,12)(21,38)(22,39)(23,40)(24,31)(25,32)(26,33)(27,34)(28,35)(29,36)(30,37), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,21)(19,22)(20,23), (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), (1,5)(2,4)(6,10)(7,9)(11,22)(12,21)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(31,39)(32,38)(33,37)(34,36) );
G=PermutationGroup([[(1,33),(2,27),(3,35),(4,29),(5,37),(6,21),(7,39),(8,23),(9,31),(10,25),(11,24),(12,32),(13,26),(14,34),(15,28),(16,36),(17,30),(18,38),(19,22),(20,40)], [(1,26),(2,14),(3,28),(4,16),(5,30),(6,18),(7,22),(8,20),(9,24),(10,12),(11,31),(13,33),(15,35),(17,37),(19,39),(21,38),(23,40),(25,32),(27,34),(29,36)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,11),(10,12),(21,38),(22,39),(23,40),(24,31),(25,32),(26,33),(27,34),(28,35),(29,36),(30,37)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,21),(19,22),(20,23)], [(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)], [(1,5),(2,4),(6,10),(7,9),(11,22),(12,21),(13,30),(14,29),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(31,39),(32,38),(33,37),(34,36)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 20 | 20 | 4 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
44 irreducible representations
| dim | 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 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C2≀C22 | D4×D5 | C24⋊2D10 |
| kernel | C24⋊2D10 | C23.1D10 | C23⋊Dic5 | C24⋊2D5 | C5×C22≀C2 | D4⋊6D10 | C2×Dic5 | C2×C20 | C22×D5 | C22×C10 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C4 | C23 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 8 |
Matrix representation of C24⋊2D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 24 | 40 |
| 0 | 0 | 1 | 17 |
| 24 | 40 | 0 | 0 |
| 1 | 17 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 17 | 1 | 0 | 0 |
| 40 | 24 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 7 |
| 0 | 0 | 34 | 7 |
| 40 | 7 | 0 | 0 |
| 34 | 7 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 34 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,24,1,0,0,40,17],[24,1,0,0,40,17,0,0,0,0,17,40,0,0,1,24],[17,40,0,0,1,24,0,0,0,0,17,40,0,0,1,24],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,40,34,0,0,7,7,40,34,0,0,7,7,0,0],[40,34,0,0,0,1,0,0,0,0,40,34,0,0,0,1] >;
C24⋊2D10 in GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_{10} % in TeX
G:=Group("C2^4:2D10"); // GroupNames label
G:=SmallGroup(320,659);
// by ID
G=gap.SmallGroup(320,659);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,570,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*b*c*d,b*c=c*b,e*b*e^-1=f*b*f=b*d=d*b,f*c*f=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations