non-abelian, soluble, monomial
Aliases: C25.D5, C24⋊Dic5, C24⋊C5⋊2C4, C2.1(C24⋊D5), (C2×C24⋊C5).C2, SmallGroup(320,1583)
Series: Derived ►Chief ►Lower central ►Upper central
| C24⋊C5 — C25.D5 |
Subgroups: 800 in 111 conjugacy classes, 7 normal (all characteristic)
C1, C2, C2 [×6], C4 [×4], C22 [×25], C5, C2×C4 [×12], C23 [×25], C10, C22⋊C4 [×12], C22×C4 [×4], C24, C24 [×6], Dic5, C2×C22⋊C4 [×6], C25, C24⋊3C4, C24⋊C5, C2×C24⋊C5, C25.D5
Quotients:
C1, C2, C4, D5, Dic5, C24⋊D5, C25.D5
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f5=1, g2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, geg-1=bc=cb, bd=db, fcf-1=gcg-1=be=eb, fbf-1=e, bg=gb, cd=dc, ce=ec, de=ed, fdf-1=bce, gdg-1=cde, fef-1=bcde, gfg-1=f-1 >
►On 20 points - transitive group 20T82
Generators in S20
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 15)(3 9)(5 20)(6 14)(7 16)(12 18)
(1 7)(2 11)(3 9)(4 10)(5 20)(6 14)(8 17)(12 18)(13 19)(15 16)
(1 15)(2 17)(3 18)(5 14)(6 20)(7 16)(8 11)(9 12)
(2 8)(4 19)(5 14)(6 20)(10 13)(11 17)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 14 7 20)(2 13 8 19)(3 12 9 18)(4 11 10 17)(5 15 6 16)
G:=sub<Sym(20)| (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,15)(3,9)(5,20)(6,14)(7,16)(12,18), (1,7)(2,11)(3,9)(4,10)(5,20)(6,14)(8,17)(12,18)(13,19)(15,16), (1,15)(2,17)(3,18)(5,14)(6,20)(7,16)(8,11)(9,12), (2,8)(4,19)(5,14)(6,20)(10,13)(11,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,7,20)(2,13,8,19)(3,12,9,18)(4,11,10,17)(5,15,6,16)>;
G:=Group( (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,15)(3,9)(5,20)(6,14)(7,16)(12,18), (1,7)(2,11)(3,9)(4,10)(5,20)(6,14)(8,17)(12,18)(13,19)(15,16), (1,15)(2,17)(3,18)(5,14)(6,20)(7,16)(8,11)(9,12), (2,8)(4,19)(5,14)(6,20)(10,13)(11,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,14,7,20)(2,13,8,19)(3,12,9,18)(4,11,10,17)(5,15,6,16) );
G=PermutationGroup([(1,7),(2,8),(3,9),(4,10),(5,6),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,15),(3,9),(5,20),(6,14),(7,16),(12,18)], [(1,7),(2,11),(3,9),(4,10),(5,20),(6,14),(8,17),(12,18),(13,19),(15,16)], [(1,15),(2,17),(3,18),(5,14),(6,20),(7,16),(8,11),(9,12)], [(2,8),(4,19),(5,14),(6,20),(10,13),(11,17)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,14,7,20),(2,13,8,19),(3,12,9,18),(4,11,10,17),(5,15,6,16)])
G:=TransitiveGroup(20,82);
►On 20 points - transitive group 20T84
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 19)(7 20)(8 16)(9 17)(10 18)
(2 12)(3 13)(7 20)(8 16)
(2 12)(4 14)(7 20)(9 17)
(2 12)(3 13)(4 14)(5 15)(7 20)(8 16)(9 17)(10 18)
(1 11)(2 12)(6 19)(7 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 9 11 17)(2 8 12 16)(3 7 13 20)(4 6 14 19)(5 10 15 18)
G:=sub<Sym(20)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,19)(7,20)(8,16)(9,17)(10,18), (2,12)(3,13)(7,20)(8,16), (2,12)(4,14)(7,20)(9,17), (2,12)(3,13)(4,14)(5,15)(7,20)(8,16)(9,17)(10,18), (1,11)(2,12)(6,19)(7,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,9,11,17)(2,8,12,16)(3,7,13,20)(4,6,14,19)(5,10,15,18)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,19)(7,20)(8,16)(9,17)(10,18), (2,12)(3,13)(7,20)(8,16), (2,12)(4,14)(7,20)(9,17), (2,12)(3,13)(4,14)(5,15)(7,20)(8,16)(9,17)(10,18), (1,11)(2,12)(6,19)(7,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,9,11,17)(2,8,12,16)(3,7,13,20)(4,6,14,19)(5,10,15,18) );
G=PermutationGroup([(1,11),(2,12),(3,13),(4,14),(5,15),(6,19),(7,20),(8,16),(9,17),(10,18)], [(2,12),(3,13),(7,20),(8,16)], [(2,12),(4,14),(7,20),(9,17)], [(2,12),(3,13),(4,14),(5,15),(7,20),(8,16),(9,17),(10,18)], [(1,11),(2,12),(6,19),(7,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,9,11,17),(2,8,12,16),(3,7,13,20),(4,6,14,19),(5,10,15,18)])
G:=TransitiveGroup(20,84);
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 40 | 2 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 23 | 40 | 0 | 0 |
| 0 | 21 | 0 | 40 | 0 |
| 0 | 26 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 |
| 1 | 19 | 0 | 1 | 0 |
| 34 | 22 | 0 | 0 | 1 |
| 1 | 39 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 |
| 0 | 20 | 0 | 1 | 0 |
| 39 | 6 | 25 | 23 | 40 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 |
| 0 | 18 | 2 | 40 | 0 |
| 0 | 22 | 7 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 9 | 1 | 0 | 0 |
| 40 | 0 | 34 | 39 | 0 |
| 0 | 0 | 0 | 7 | 1 |
| 24 | 5 | 30 | 35 | 32 |
| 0 | 23 | 40 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 1 | 39 | 0 | 0 | 0 |
| 33 | 12 | 31 | 1 | 32 |
| 39 | 3 | 18 | 23 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,2,1,23,21,26,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,1,34,0,40,18,19,22,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,39,39,40,18,20,6,0,0,1,0,25,0,0,0,1,23,0,0,0,0,40],[40,0,0,0,0,0,40,18,18,22,0,0,1,2,7,0,0,0,40,0,0,0,0,0,40],[0,0,40,0,24,0,9,0,0,5,1,1,34,0,30,0,0,39,7,35,0,0,0,1,32],[0,0,1,33,39,23,32,39,12,3,40,0,0,31,18,0,0,0,1,23,0,0,0,32,40] >;
Character table of C25.D5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | 10B | |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 32 | 32 | 32 | 32 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | i | i | -i | -i | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | -i | -i | i | i | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ8 | 2 | -2 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | symplectic lifted from Dic5, Schur index 2 |
| ρ9 | 5 | 5 | -3 | 1 | 1 | 1 | -3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ10 | 5 | 5 | -3 | 1 | 1 | 1 | -3 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ11 | 5 | 5 | 1 | 1 | -3 | -3 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ12 | 5 | 5 | 1 | -3 | 1 | 1 | 1 | -3 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ13 | 5 | 5 | 1 | -3 | 1 | 1 | 1 | -3 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ14 | 5 | 5 | 1 | 1 | -3 | -3 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ15 | 5 | -5 | 1 | -3 | 1 | -1 | -1 | 3 | -i | i | -i | i | i | -i | i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ16 | 5 | -5 | 1 | 1 | -3 | 3 | -1 | -1 | -i | -i | i | i | i | i | -i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ17 | 5 | -5 | -3 | 1 | 1 | -1 | 3 | -1 | i | i | i | -i | i | -i | -i | -i | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 5 | -5 | 1 | -3 | 1 | -1 | -1 | 3 | i | -i | i | -i | -i | i | -i | i | 0 | 0 | 0 | 0 | complex faithful |
| ρ19 | 5 | -5 | 1 | 1 | -3 | 3 | -1 | -1 | i | i | -i | -i | -i | -i | i | i | 0 | 0 | 0 | 0 | complex faithful |
| ρ20 | 5 | -5 | -3 | 1 | 1 | -1 | 3 | -1 | -i | -i | -i | i | -i | i | i | i | 0 | 0 | 0 | 0 | complex faithful |
In GAP, Magma, Sage, TeX
C_2^5.D_5
% in TeX
G:=Group("C2^5.D5"); // GroupNames label
G:=SmallGroup(320,1583);
// by ID
G=gap.SmallGroup(320,1583);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,338,1683,437,1068,9245,2539,4906,265]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^5=1,g^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,g*e*g^-1=b*c=c*b,b*d=d*b,f*c*f^-1=g*c*g^-1=b*e=e*b,f*b*f^-1=e,b*g=g*b,c*d=d*c,c*e=e*c,d*e=e*d,f*d*f^-1=b*c*e,g*d*g^-1=c*d*e,f*e*f^-1=b*c*d*e,g*f*g^-1=f^-1>;
// generators/relations
×
⋊
ℤ
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