Aliases: 2- (1+4).C10, C4.(C24⋊C5), C2.C25⋊C5, 2- (1+4)⋊C5⋊2C2, C2.3(C2×C24⋊C5), SmallGroup(320,1586)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — 2- (1+4) — 2- (1+4)⋊C5 — 2- (1+4).C10 |
| 2- (1+4) — 2- (1+4).C10 |
Subgroups: 515 in 90 conjugacy classes, 7 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×6], C5, C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C10, C22×C4 [×3], C2×D4 [×9], C2×Q8 [×3], C4○D4 [×16], C20, C2×C4○D4 [×3], 2+ (1+4) [×2], 2- (1+4), 2- (1+4), C2.C25, 2- (1+4)⋊C5, 2- (1+4).C10
Quotients:
C1, C2, C5, C10, C24⋊C5, C2×C24⋊C5, 2- (1+4).C10
Generators and relations
G = < a,b,c,d,e | a4=b2=1, c2=d2=e10=a2, bab=a-1, ebe-1=ac=ca, ad=da, eae-1=cd, bc=cb, bd=db, dcd-1=a2c, ece-1=a-1bcd, ede-1=a >
►On 64 points
Generators in S64
(1 57 3 47)(2 62 4 52)(5 63 15 53)(6 18 16 8)(7 40 17 30)(9 51 19 61)(10 48 20 58)(11 23 21 13)(12 25 22 35)(14 56 24 46)(26 49 36 59)(27 33 37 43)(28 32 38 42)(29 50 39 60)(31 54 41 64)(34 55 44 45)
(1 13)(2 18)(3 23)(4 8)(5 59)(6 62)(7 38)(9 34)(10 64)(11 47)(12 43)(14 39)(15 49)(16 52)(17 28)(19 44)(20 54)(21 57)(22 33)(24 29)(25 37)(26 53)(27 35)(30 42)(31 58)(32 40)(36 63)(41 48)(45 51)(46 60)(50 56)(55 61)
(1 40 3 30)(2 25 4 35)(5 19 15 9)(6 33 16 43)(7 57 17 47)(8 27 18 37)(10 24 20 14)(11 38 21 28)(12 62 22 52)(13 32 23 42)(26 45 36 55)(29 54 39 64)(31 50 41 60)(34 59 44 49)(46 58 56 48)(51 63 61 53)
(1 53 3 63)(2 58 4 48)(5 47 15 57)(6 64 16 54)(7 19 17 9)(8 41 18 31)(10 52 20 62)(11 49 21 59)(12 24 22 14)(13 26 23 36)(25 46 35 56)(27 50 37 60)(28 34 38 44)(29 33 39 43)(30 51 40 61)(32 55 42 45)
(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 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)
G:=sub<Sym(64)| (1,57,3,47)(2,62,4,52)(5,63,15,53)(6,18,16,8)(7,40,17,30)(9,51,19,61)(10,48,20,58)(11,23,21,13)(12,25,22,35)(14,56,24,46)(26,49,36,59)(27,33,37,43)(28,32,38,42)(29,50,39,60)(31,54,41,64)(34,55,44,45), (1,13)(2,18)(3,23)(4,8)(5,59)(6,62)(7,38)(9,34)(10,64)(11,47)(12,43)(14,39)(15,49)(16,52)(17,28)(19,44)(20,54)(21,57)(22,33)(24,29)(25,37)(26,53)(27,35)(30,42)(31,58)(32,40)(36,63)(41,48)(45,51)(46,60)(50,56)(55,61), (1,40,3,30)(2,25,4,35)(5,19,15,9)(6,33,16,43)(7,57,17,47)(8,27,18,37)(10,24,20,14)(11,38,21,28)(12,62,22,52)(13,32,23,42)(26,45,36,55)(29,54,39,64)(31,50,41,60)(34,59,44,49)(46,58,56,48)(51,63,61,53), (1,53,3,63)(2,58,4,48)(5,47,15,57)(6,64,16,54)(7,19,17,9)(8,41,18,31)(10,52,20,62)(11,49,21,59)(12,24,22,14)(13,26,23,36)(25,46,35,56)(27,50,37,60)(28,34,38,44)(29,33,39,43)(30,51,40,61)(32,55,42,45), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;
G:=Group( (1,57,3,47)(2,62,4,52)(5,63,15,53)(6,18,16,8)(7,40,17,30)(9,51,19,61)(10,48,20,58)(11,23,21,13)(12,25,22,35)(14,56,24,46)(26,49,36,59)(27,33,37,43)(28,32,38,42)(29,50,39,60)(31,54,41,64)(34,55,44,45), (1,13)(2,18)(3,23)(4,8)(5,59)(6,62)(7,38)(9,34)(10,64)(11,47)(12,43)(14,39)(15,49)(16,52)(17,28)(19,44)(20,54)(21,57)(22,33)(24,29)(25,37)(26,53)(27,35)(30,42)(31,58)(32,40)(36,63)(41,48)(45,51)(46,60)(50,56)(55,61), (1,40,3,30)(2,25,4,35)(5,19,15,9)(6,33,16,43)(7,57,17,47)(8,27,18,37)(10,24,20,14)(11,38,21,28)(12,62,22,52)(13,32,23,42)(26,45,36,55)(29,54,39,64)(31,50,41,60)(34,59,44,49)(46,58,56,48)(51,63,61,53), (1,53,3,63)(2,58,4,48)(5,47,15,57)(6,64,16,54)(7,19,17,9)(8,41,18,31)(10,52,20,62)(11,49,21,59)(12,24,22,14)(13,26,23,36)(25,46,35,56)(27,50,37,60)(28,34,38,44)(29,33,39,43)(30,51,40,61)(32,55,42,45), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );
G=PermutationGroup([(1,57,3,47),(2,62,4,52),(5,63,15,53),(6,18,16,8),(7,40,17,30),(9,51,19,61),(10,48,20,58),(11,23,21,13),(12,25,22,35),(14,56,24,46),(26,49,36,59),(27,33,37,43),(28,32,38,42),(29,50,39,60),(31,54,41,64),(34,55,44,45)], [(1,13),(2,18),(3,23),(4,8),(5,59),(6,62),(7,38),(9,34),(10,64),(11,47),(12,43),(14,39),(15,49),(16,52),(17,28),(19,44),(20,54),(21,57),(22,33),(24,29),(25,37),(26,53),(27,35),(30,42),(31,58),(32,40),(36,63),(41,48),(45,51),(46,60),(50,56),(55,61)], [(1,40,3,30),(2,25,4,35),(5,19,15,9),(6,33,16,43),(7,57,17,47),(8,27,18,37),(10,24,20,14),(11,38,21,28),(12,62,22,52),(13,32,23,42),(26,45,36,55),(29,54,39,64),(31,50,41,60),(34,59,44,49),(46,58,56,48),(51,63,61,53)], [(1,53,3,63),(2,58,4,48),(5,47,15,57),(6,64,16,54),(7,19,17,9),(8,41,18,31),(10,52,20,62),(11,49,21,59),(12,24,22,14),(13,26,23,36),(25,46,35,56),(27,50,37,60),(28,34,38,44),(29,33,39,43),(30,51,40,61),(32,55,42,45)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)])
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 3 | 0 | 3 | 1 |
| 4 | 2 | 2 | 4 |
| 1 | 1 | 3 | 0 |
| 2 | 2 | 2 | 2 |
| 4 | 0 | 0 | 4 |
| 4 | 0 | 4 | 4 |
| 1 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 1 | 4 | 0 |
| 2 | 3 | 4 | 1 |
| 2 | 0 | 2 | 1 |
| 2 | 3 | 2 | 3 |
| 2 | 4 | 3 | 1 |
| 1 | 4 | 2 | 4 |
| 0 | 4 | 0 | 1 |
| 1 | 4 | 1 | 4 |
| 3 | 2 | 0 | 2 |
| 0 | 1 | 3 | 3 |
| 0 | 4 | 0 | 2 |
| 0 | 3 | 0 | 3 |
G:=sub<GL(4,GF(5))| [3,4,1,2,0,2,1,2,3,2,3,2,1,4,0,2],[4,4,1,0,0,0,4,0,0,4,0,0,4,4,0,1],[2,2,2,2,1,3,0,3,4,4,2,2,0,1,1,3],[2,1,0,1,4,4,4,4,3,2,0,1,1,4,1,4],[3,0,0,0,2,1,4,3,0,3,0,0,2,3,2,3] >;
Character table of 2- (1+4).C10
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 10 | 10 | 10 | 1 | 1 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ54 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | ζ5 | linear of order 5 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ53 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | ζ52 | linear of order 5 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ52 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | ζ53 | linear of order 5 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | -ζ5 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | -ζ54 | linear of order 10 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ5 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | ζ54 | linear of order 5 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | -ζ53 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | -ζ52 | linear of order 10 |
| ρ9 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | -ζ54 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | -ζ5 | linear of order 10 |
| ρ10 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | -ζ52 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | -ζ53 | linear of order 10 |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | i | -i | i | -i | -i | complex faithful |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | -i | i | -i | i | i | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ52 | -ζ54 | -ζ53 | -ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ4ζ54 | ζ4ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ54 | ζ43ζ5 | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ54 | -ζ53 | -ζ5 | -ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ4ζ53 | ζ4ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ53 | ζ43ζ52 | complex faithful |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ53 | -ζ5 | -ζ52 | -ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ43ζ5 | ζ43ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ5 | ζ4ζ54 | complex faithful |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ54 | -ζ53 | -ζ5 | -ζ52 | ζ54 | ζ53 | ζ5 | ζ52 | ζ43ζ53 | ζ43ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ53 | ζ4ζ52 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ5 | -ζ52 | -ζ54 | -ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ43ζ52 | ζ43ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ52 | ζ4ζ53 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ5 | -ζ52 | -ζ54 | -ζ53 | ζ5 | ζ52 | ζ54 | ζ53 | ζ4ζ52 | ζ4ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ52 | ζ43ζ53 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ53 | -ζ5 | -ζ52 | -ζ54 | ζ53 | ζ5 | ζ52 | ζ54 | ζ4ζ5 | ζ4ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ5 | ζ43ζ54 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 4i | 4i | 0 | 0 | 0 | -ζ52 | -ζ54 | -ζ53 | -ζ5 | ζ52 | ζ54 | ζ53 | ζ5 | ζ43ζ54 | ζ43ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ54 | ζ4ζ5 | complex faithful |
| ρ21 | 5 | 5 | 1 | -1 | 3 | -5 | -5 | -1 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C24⋊C5 |
| ρ22 | 5 | 5 | 1 | -3 | 1 | 5 | 5 | 1 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
| ρ23 | 5 | 5 | -3 | 1 | 1 | 5 | 5 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
| ρ24 | 5 | 5 | 1 | 3 | -1 | -5 | -5 | -1 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C24⋊C5 |
| ρ25 | 5 | 5 | -3 | -1 | -1 | -5 | -5 | 3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C24⋊C5 |
| ρ26 | 5 | 5 | 1 | 1 | -3 | 5 | 5 | 1 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
In GAP, Magma, Sage, TeX
2_-^{(1+4)}.C_{10} % in TeX
G:=Group("ES-(2,2).C10"); // GroupNames label
G:=SmallGroup(320,1586);
// by ID
G=gap.SmallGroup(320,1586);
# by ID
G:=PCGroup([7,-2,-5,-2,2,2,2,-2,1120,849,1270,521,248,1936,718,375,172,3162,1027]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=1,c^2=d^2=e^10=a^2,b*a*b=a^-1,e*b*e^-1=a*c=c*a,a*d=d*a,e*a*e^-1=c*d,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=a^-1*b*c*d,e*d*e^-1=a>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀