metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊2D5, C23.25D10, C5⋊3C22≀C2, (C2×C10)⋊8D4, (C23×C10)⋊3C2, C10.63(C2×D4), C22⋊3(C5⋊D4), C23.D5⋊13C2, (C2×C10).61C23, (C2×Dic5)⋊3C22, (C22×D5)⋊2C22, C22.66(C22×D5), (C22×C10).42C22, (C2×C5⋊D4)⋊8C2, C2.26(C2×C5⋊D4), SmallGroup(160,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊2D5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 368 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, D5, C10, C10, C22⋊C4, C2×D4, C24, Dic5, D10, C2×C10, C2×C10, C2×C10, C22≀C2, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C22×C10, C23.D5, C2×C5⋊D4, C23×C10, C24⋊2D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C5⋊D4, C22×D5, C2×C5⋊D4, C24⋊2D5
►On 40 points
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 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 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 25)(2 24)(3 23)(4 22)(5 21)(6 28)(7 27)(8 26)(9 30)(10 29)(11 33)(12 32)(13 31)(14 35)(15 34)(16 38)(17 37)(18 36)(19 40)(20 39)
G:=sub<Sym(40)| (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,9)(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), (1,2,3,4,5)(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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,28)(7,27)(8,26)(9,30)(10,29)(11,33)(12,32)(13,31)(14,35)(15,34)(16,38)(17,37)(18,36)(19,40)(20,39)>;
G:=Group( (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,9)(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), (1,2,3,4,5)(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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,28)(7,27)(8,26)(9,30)(10,29)(11,33)(12,32)(13,31)(14,35)(15,34)(16,38)(17,37)(18,36)(19,40)(20,39) );
G=PermutationGroup([[(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(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)], [(1,9),(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)], [(1,2,3,4,5),(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,28),(7,27),(8,26),(9,30),(10,29),(11,33),(12,32),(13,31),(14,35),(15,34),(16,38),(17,37),(18,36),(19,40),(20,39)]])
C24⋊2D5 is a maximal subgroup of
C24⋊2F5 C24⋊2D10 C24.27D10 C24.30D10 C24.31D10 C24.56D10 D5×C22≀C2 C24⋊3D10 C24.34D10 C24.35D10 C24⋊5D10 C24.36D10 C24.72D10 D4×C5⋊D4 C24⋊8D10 C24.41D10 C24.42D10 (C22×D5)⋊A4 C15⋊C22≀C2 (C2×C10)⋊11D12 C24⋊5D15 C24⋊2D15 C24⋊4D15
C24⋊2D5 is a maximal quotient of
C24.62D10 C24.65D10 (C2×C10)⋊8D8 (C5×D4).31D4 C24.20D10 C24.21D10 (C5×Q8)⋊13D4 (C2×C10)⋊8Q16 C10.C22≀C2 (C22×D5)⋊Q8 (C5×D4)⋊14D4 (C5×D4).32D4 2+ 1+4⋊D5 2+ 1+4.D5 2+ 1+4.2D5 2+ 1+4⋊2D5 2- 1+4⋊2D5 2- 1+4.2D5 C25.2D5 C15⋊C22≀C2 (C2×C10)⋊11D12 C24⋊5D15
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 4A | 4B | 4C | 5A | 5B | 10A | ··· | 10AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D10 | C5⋊D4 |
| kernel | C24⋊2D5 | C23.D5 | C2×C5⋊D4 | C23×C10 | C2×C10 | C24 | C23 | C22 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 6 | 24 |
Matrix representation of C24⋊2D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 16 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 18 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 |
| 0 | 0 | 10 | 0 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[16,0,0,0,0,18,0,0,0,0,10,0,0,0,0,37],[0,16,0,0,18,0,0,0,0,0,0,10,0,0,37,0] >;
C24⋊2D5 in GAP, Magma, Sage, TeX
C_2^4\rtimes_2D_5
% in TeX
G:=Group("C2^4:2D5"); // GroupNames label
G:=SmallGroup(160,174);
// by ID
G=gap.SmallGroup(160,174);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,218,4613]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations