metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊Dic5, C23.2D10, (C2×C4)⋊Dic5, (C2×C20)⋊6C4, C5⋊4(C23⋊C4), (C2×D4).3D5, (C2×C10).2D4, (C22×C10)⋊2C4, (D4×C10).6C2, C23.D5⋊2C2, C22.2(C5⋊D4), C2.5(C23.D5), C22.3(C2×Dic5), C10.26(C22⋊C4), (C22×C10).6C22, (C2×C10).49(C2×C4), SmallGroup(160,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊Dic5
G = < a,b,c,d,e | a2=b2=c2=d10=1, e2=d5, ab=ba, dad-1=ac=ca, eae-1=abc, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Smallest permutation representation of C23⋊Dic5
►On 40 points
(1 14)(2 35)(3 16)(4 37)(5 18)(6 39)(7 20)(8 31)(9 12)(10 33)(11 30)(13 22)(15 24)(17 26)(19 28)(21 32)(23 34)(25 36)(27 38)(29 40)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 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 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)
G:=sub<Sym(40)| (1,14)(2,35)(3,16)(4,37)(5,18)(6,39)(7,20)(8,31)(9,12)(10,33)(11,30)(13,22)(15,24)(17,26)(19,28)(21,32)(23,34)(25,36)(27,38)(29,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (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), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)>;
G:=Group( (1,14)(2,35)(3,16)(4,37)(5,18)(6,39)(7,20)(8,31)(9,12)(10,33)(11,30)(13,22)(15,24)(17,26)(19,28)(21,32)(23,34)(25,36)(27,38)(29,40), (1,6)(2,7)(3,8)(4,9)(5,10)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(21,26)(22,27)(23,28)(24,29)(25,30), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40), (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), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37) );
G=PermutationGroup([[(1,14),(2,35),(3,16),(4,37),(5,18),(6,39),(7,20),(8,31),(9,12),(10,33),(11,30),(13,22),(15,24),(17,26),(19,28),(21,32),(23,34),(25,36),(27,38),(29,40)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,36),(12,37),(13,38),(14,39),(15,40),(16,31),(17,32),(18,33),(19,34),(20,35),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,23),(2,24),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,21),(10,22),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)], [(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)], [(1,14,6,19),(2,13,7,18),(3,12,8,17),(4,11,9,16),(5,20,10,15),(21,36,26,31),(22,35,27,40),(23,34,28,39),(24,33,29,38),(25,32,30,37)]])
C23⋊Dic5 is a maximal subgroup of
C23.D20 C23.2D20 C23.3D20 C23.4D20 C24⋊2Dic5 (C22×C20)⋊C4 C42⋊Dic5 C42⋊3Dic5 C23⋊C4⋊5D5 D5×C23⋊C4 C24⋊2D10 C22⋊C4⋊D10 (D4×C10)⋊22C4 2+ 1+4.2D5 2+ 1+4⋊2D5 C15⋊8(C23⋊C4) C23.7D30
C23⋊Dic5 is a maximal quotient of
C24.Dic5 C24.D10 (C2×C20)⋊C8 C24⋊2Dic5 C4⋊C4⋊Dic5 C10.29C4≀C2 (C22×C20)⋊C4 C42⋊Dic5 C42.Dic5 C42⋊3Dic5 C42.3Dic5 C15⋊8(C23⋊C4) C23.7D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D5 | Dic5 | Dic5 | D10 | C5⋊D4 | C23⋊C4 | C23⋊Dic5 |
| kernel | C23⋊Dic5 | C23.D5 | D4×C10 | C2×C20 | C22×C10 | C2×C10 | C2×D4 | C2×C4 | C23 | C23 | C22 | C5 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 4 |
Matrix representation of C23⋊Dic5 ►in GL4(𝔽41) generated by
| 24 | 1 | 38 | 23 |
| 40 | 17 | 36 | 23 |
| 0 | 0 | 18 | 40 |
| 0 | 0 | 36 | 23 |
| 24 | 1 | 0 | 0 |
| 40 | 17 | 0 | 0 |
| 0 | 0 | 23 | 1 |
| 0 | 0 | 5 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 17 | 0 | 0 |
| 24 | 3 | 0 | 0 |
| 40 | 33 | 18 | 24 |
| 7 | 6 | 38 | 21 |
| 40 | 0 | 36 | 40 |
| 7 | 1 | 35 | 6 |
| 11 | 29 | 35 | 40 |
| 4 | 18 | 35 | 6 |
G:=sub<GL(4,GF(41))| [24,40,0,0,1,17,0,0,38,36,18,36,23,23,40,23],[24,40,0,0,1,17,0,0,0,0,23,5,0,0,1,18],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,24,40,7,17,3,33,6,0,0,18,38,0,0,24,21],[40,7,11,4,0,1,29,18,36,35,35,35,40,6,40,6] >;
C23⋊Dic5 in GAP, Magma, Sage, TeX
C_2^3\rtimes {\rm Dic}_5 % in TeX
G:=Group("C2^3:Dic5"); // GroupNames label
G:=SmallGroup(160,41);
// by ID
G=gap.SmallGroup(160,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,188,579,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^10=1,e^2=d^5,a*b=b*a,d*a*d^-1=a*c=c*a,e*a*e^-1=a*b*c,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export