Aliases: 2- (1+4)⋊D5, 2- (1+4)⋊C5⋊C2, C2.3(C24⋊D5), SmallGroup(320,1582)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — 2- (1+4)⋊D5 |
| C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — 2- (1+4)⋊D5 |
| 2- (1+4)⋊C5 — 2- (1+4)⋊D5 |
Subgroups: 455 in 52 conjugacy classes, 5 normal (all characteristic)
C1, C2, C2 [×2], C4 [×3], C22 [×4], C5, C8 [×2], C2×C4 [×4], D4 [×4], Q8 [×2], C23, D5 [×2], C10, C42, C22⋊C4 [×2], M4(2) [×2], D8 [×2], SD16 [×2], C2×D4, C2×Q8, C4○D4 [×2], D10, C4.10D4, C4≀C2 [×2], C4.4D4, C8⋊C22 [×2], 2- (1+4), D4.8D4, 2- (1+4)⋊C5, 2- (1+4)⋊D5
Quotients:
C1, C2, D5, C24⋊D5, 2- (1+4)⋊D5
Generators and relations
G = < a,b,c,d,e,f | a4=b2=e5=f2=1, c2=d2=a2, bab=a-1, ac=ca, ebe-1=ad=da, eae-1=fdf=a-1bd, faf=ede-1=bcd, bc=cb, bd=db, fbf=acd, dcd-1=a2c, ece-1=a2bc, fcf=abc, fef=e-1 >
►On 32 points
Generators in S32
(1 31 2 11)(3 20 19 4)(5 9 21 29)(6 23 22 16)(7 8 18 28)(10 25 30 13)(12 14 32 26)(15 24 27 17)
(1 19)(2 3)(4 11)(5 14)(6 24)(7 30)(8 25)(9 12)(10 18)(13 28)(15 23)(16 27)(17 22)(20 31)(21 26)(29 32)
(1 16 2 23)(3 15 19 27)(4 17 20 24)(5 7 21 18)(6 11 22 31)(8 29 28 9)(10 14 30 26)(12 25 32 13)
(1 8 2 28)(3 13 19 25)(4 30 20 10)(5 6 21 22)(7 31 18 11)(9 23 29 16)(12 15 32 27)(14 24 26 17)
(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)
(1 2)(3 21)(4 20)(5 19)(6 18)(7 22)(8 10)(11 12)(13 15)(16 17)(23 24)(25 27)(28 30)(31 32)
G:=sub<Sym(32)| (1,31,2,11)(3,20,19,4)(5,9,21,29)(6,23,22,16)(7,8,18,28)(10,25,30,13)(12,14,32,26)(15,24,27,17), (1,19)(2,3)(4,11)(5,14)(6,24)(7,30)(8,25)(9,12)(10,18)(13,28)(15,23)(16,27)(17,22)(20,31)(21,26)(29,32), (1,16,2,23)(3,15,19,27)(4,17,20,24)(5,7,21,18)(6,11,22,31)(8,29,28,9)(10,14,30,26)(12,25,32,13), (1,8,2,28)(3,13,19,25)(4,30,20,10)(5,6,21,22)(7,31,18,11)(9,23,29,16)(12,15,32,27)(14,24,26,17), (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), (1,2)(3,21)(4,20)(5,19)(6,18)(7,22)(8,10)(11,12)(13,15)(16,17)(23,24)(25,27)(28,30)(31,32)>;
G:=Group( (1,31,2,11)(3,20,19,4)(5,9,21,29)(6,23,22,16)(7,8,18,28)(10,25,30,13)(12,14,32,26)(15,24,27,17), (1,19)(2,3)(4,11)(5,14)(6,24)(7,30)(8,25)(9,12)(10,18)(13,28)(15,23)(16,27)(17,22)(20,31)(21,26)(29,32), (1,16,2,23)(3,15,19,27)(4,17,20,24)(5,7,21,18)(6,11,22,31)(8,29,28,9)(10,14,30,26)(12,25,32,13), (1,8,2,28)(3,13,19,25)(4,30,20,10)(5,6,21,22)(7,31,18,11)(9,23,29,16)(12,15,32,27)(14,24,26,17), (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), (1,2)(3,21)(4,20)(5,19)(6,18)(7,22)(8,10)(11,12)(13,15)(16,17)(23,24)(25,27)(28,30)(31,32) );
G=PermutationGroup([(1,31,2,11),(3,20,19,4),(5,9,21,29),(6,23,22,16),(7,8,18,28),(10,25,30,13),(12,14,32,26),(15,24,27,17)], [(1,19),(2,3),(4,11),(5,14),(6,24),(7,30),(8,25),(9,12),(10,18),(13,28),(15,23),(16,27),(17,22),(20,31),(21,26),(29,32)], [(1,16,2,23),(3,15,19,27),(4,17,20,24),(5,7,21,18),(6,11,22,31),(8,29,28,9),(10,14,30,26),(12,25,32,13)], [(1,8,2,28),(3,13,19,25),(4,30,20,10),(5,6,21,22),(7,31,18,11),(9,23,29,16),(12,15,32,27),(14,24,26,17)], [(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)], [(1,2),(3,21),(4,20),(5,19),(6,18),(7,22),(8,10),(11,12),(13,15),(16,17),(23,24),(25,27),(28,30),(31,32)])
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 1 | 0 | 0 | 2 |
| 0 | 0 | 4 | 3 |
| 2 | 1 | 0 | 2 |
| 4 | 0 | 0 | 4 |
| 3 | 2 | 3 | 4 |
| 2 | 1 | 4 | 0 |
| 3 | 4 | 0 | 3 |
| 1 | 0 | 2 | 1 |
| 2 | 0 | 3 | 0 |
| 1 | 0 | 2 | 1 |
| 0 | 0 | 3 | 0 |
| 3 | 4 | 1 | 0 |
| 2 | 1 | 3 | 1 |
| 2 | 3 | 0 | 3 |
| 1 | 3 | 2 | 4 |
| 0 | 1 | 3 | 3 |
| 1 | 2 | 2 | 2 |
| 0 | 0 | 1 | 2 |
| 0 | 1 | 0 | 3 |
| 0 | 1 | 3 | 3 |
| 1 | 3 | 3 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 2 | 4 | 1 |
G:=sub<GL(4,GF(5))| [1,0,2,4,0,0,1,0,0,4,0,0,2,3,2,4],[3,2,3,1,2,1,4,0,3,4,0,2,4,0,3,1],[2,1,0,3,0,0,0,4,3,2,3,1,0,1,0,0],[2,2,1,0,1,3,3,1,3,0,2,3,1,3,4,3],[1,0,0,0,2,0,1,1,2,1,0,3,2,2,3,3],[1,0,0,0,3,4,0,2,3,0,4,4,0,0,0,1] >;
Character table of 2- (1+4)⋊D5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 10A | 10B | |
| size | 1 | 1 | 10 | 40 | 10 | 10 | 20 | 20 | 32 | 32 | 40 | 40 | 32 | 32 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ4 | 2 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ5 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | 2i | -1 | -1 | 0 | 0 | 1 | 1 | complex faithful |
| ρ6 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | 2i | -1 | -1 | 0 | 0 | 1 | 1 | complex faithful |
| ρ7 | 5 | 5 | -3 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ8 | 5 | 5 | 1 | -1 | -3 | 1 | 1 | 1 | 0 | 0 | -1 | 1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ9 | 5 | 5 | 1 | -1 | 1 | -3 | 1 | 1 | 0 | 0 | 1 | -1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ10 | 5 | 5 | 1 | 1 | -3 | 1 | -1 | -1 | 0 | 0 | 1 | -1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ11 | 5 | 5 | 1 | 1 | 1 | -3 | -1 | -1 | 0 | 0 | -1 | 1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ12 | 5 | 5 | -3 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from C24⋊D5 |
| ρ13 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ14 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
In GAP, Magma, Sage, TeX
2_-^{(1+4)}\rtimes D_5 % in TeX
G:=Group("ES-(2,2):D5"); // GroupNames label
G:=SmallGroup(320,1582);
// by ID
G=gap.SmallGroup(320,1582);
# by ID
G:=PCGroup([7,-2,-5,-2,2,2,2,-2,113,632,324,5043,850,521,248,3854,2111,718,375,172,2105,3582,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=e^5=f^2=1,c^2=d^2=a^2,b*a*b=a^-1,a*c=c*a,e*b*e^-1=a*d=d*a,e*a*e^-1=f*d*f=a^-1*b*d,f*a*f=e*d*e^-1=b*c*d,b*c=c*b,b*d=d*b,f*b*f=a*c*d,d*c*d^-1=a^2*c,e*c*e^-1=a^2*b*c,f*c*f=a*b*c,f*e*f=e^-1>;
// generators/relations
×
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ℤ
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