direct product, non-abelian, soluble
Aliases: C2×2- 1+4⋊C5, 2- 1+4⋊C10, (C2×2- 1+4)⋊C5, C22.1(C24⋊C5), C2.2(C2×C24⋊C5), SmallGroup(320,1585)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C2 — 2- 1+4 — C2×2- 1+4⋊C5 |
C1 — C2 — 2- 1+4 — 2- 1+4⋊C5 — C2×2- 1+4⋊C5 |
2- 1+4 — C2×2- 1+4⋊C5 |
Generators and relations for C2×2- 1+4⋊C5
G = < a,b,c,d,e,f | a2=b4=c2=f5=1, d2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2cde, cd=dc, ce=ec, fcf-1=bc, ede-1=b2d, fdf-1=bcd, fef-1=de >
Subgroups: 499 in 92 conjugacy classes, 9 normal (7 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, Q8, C23, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C10, C22×Q8, C2×C4○D4, 2- 1+4, 2- 1+4, C2×2- 1+4, 2- 1+4⋊C5, C2×2- 1+4⋊C5
Quotients: C1, C2, C5, C10, C24⋊C5, 2- 1+4⋊C5, C2×C24⋊C5, C2×2- 1+4⋊C5
Character table of C2×2- 1+4⋊C5
class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | |
size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 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 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | ζ5 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | ζ54 | linear of order 5 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | ζ53 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | ζ52 | linear of order 5 |
ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | ζ53 | ζ54 | ζ5 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | -ζ5 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | -ζ54 | linear of order 10 |
ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | -ζ54 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | -ζ5 | linear of order 10 |
ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | -ζ52 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | -ζ53 | linear of order 10 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ5 | ζ53 | ζ52 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | ζ52 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | ζ53 | linear of order 5 |
ρ9 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | ζ54 | ζ52 | ζ53 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | -ζ53 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | -ζ52 | linear of order 10 |
ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ52 | ζ5 | ζ54 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | ζ54 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | ζ5 | linear of order 5 |
ρ11 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | symplectic lifted from 2- 1+4⋊C5, Schur index 2 |
ρ12 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from 2- 1+4⋊C5, Schur index 2 |
ρ13 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | ζ54 | -ζ54 | ζ53 | -ζ53 | ζ52 | -ζ52 | ζ5 | -ζ5 | complex lifted from 2- 1+4⋊C5 |
ρ14 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ5 | -ζ54 | -ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | -ζ54 | ζ54 | -ζ53 | ζ53 | -ζ52 | ζ52 | -ζ5 | ζ5 | complex lifted from 2- 1+4⋊C5 |
ρ15 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | -ζ52 | ζ52 | -ζ54 | ζ54 | -ζ5 | ζ5 | -ζ53 | ζ53 | complex lifted from 2- 1+4⋊C5 |
ρ16 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ53 | -ζ52 | -ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | ζ52 | -ζ52 | ζ54 | -ζ54 | ζ5 | -ζ5 | ζ53 | -ζ53 | complex lifted from 2- 1+4⋊C5 |
ρ17 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | -ζ5 | ζ5 | -ζ52 | ζ52 | -ζ53 | ζ53 | -ζ54 | ζ54 | complex lifted from 2- 1+4⋊C5 |
ρ18 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ54 | -ζ5 | -ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | ζ5 | -ζ5 | ζ52 | -ζ52 | ζ53 | -ζ53 | ζ54 | -ζ54 | complex lifted from 2- 1+4⋊C5 |
ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | ζ53 | -ζ53 | ζ5 | -ζ5 | ζ54 | -ζ54 | ζ52 | -ζ52 | complex lifted from 2- 1+4⋊C5 |
ρ20 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ52 | -ζ53 | -ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | -ζ53 | ζ53 | -ζ5 | ζ5 | -ζ54 | ζ54 | -ζ52 | ζ52 | complex lifted from 2- 1+4⋊C5 |
ρ21 | 5 | -5 | 5 | -5 | -1 | 1 | -3 | 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 |
ρ22 | 5 | 5 | 5 | 5 | 1 | 1 | -3 | 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 | 5 | -5 | 3 | -3 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C24⋊C5 |
ρ24 | 5 | 5 | 5 | 5 | -3 | -3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
ρ25 | 5 | -5 | 5 | -5 | -1 | 1 | 1 | -3 | 3 | -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 | 5 | 5 | 1 | 1 | 1 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C5 |
(1 3)(2 4)(5 22)(6 23)(7 24)(8 20)(9 21)(10 44)(11 40)(12 41)(13 42)(14 43)(15 50)(16 51)(17 52)(18 53)(19 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 61)(31 62)(32 63)(33 64)(34 60)(35 46)(36 47)(37 48)(38 49)(39 45)
(1 55 4 53)(2 18 3 25)(5 29 46 17)(6 19 47 26)(7 9 48 45)(8 34 49 44)(10 20 60 38)(11 57 61 50)(12 13 62 63)(14 58 64 51)(15 40 27 30)(16 43 28 33)(21 37 39 24)(22 59 35 52)(23 54 36 56)(31 32 41 42)
(1 63)(2 42)(3 32)(4 13)(5 40)(6 16)(7 34)(8 9)(10 37)(11 22)(12 53)(14 56)(15 29)(17 27)(18 41)(19 33)(20 21)(23 51)(24 60)(25 31)(26 43)(28 47)(30 46)(35 61)(36 58)(38 39)(44 48)(45 49)(50 59)(52 57)(54 64)(55 62)
(1 20 4 38)(2 49 3 8)(5 26 46 19)(6 17 47 29)(7 41 48 31)(9 42 45 32)(10 55 60 53)(11 14 61 64)(12 37 62 24)(13 39 63 21)(15 16 27 28)(18 44 25 34)(22 56 35 54)(23 52 36 59)(30 33 40 43)(50 51 57 58)
(1 58 4 51)(2 16 3 28)(5 7 46 48)(6 32 47 42)(8 27 49 15)(9 17 45 29)(10 11 60 61)(12 56 62 54)(13 23 63 36)(14 55 64 53)(18 43 25 33)(19 41 26 31)(20 57 38 50)(21 52 39 59)(22 24 35 37)(30 44 40 34)
(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)| (1,3)(2,4)(5,22)(6,23)(7,24)(8,20)(9,21)(10,44)(11,40)(12,41)(13,42)(14,43)(15,50)(16,51)(17,52)(18,53)(19,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,61)(31,62)(32,63)(33,64)(34,60)(35,46)(36,47)(37,48)(38,49)(39,45), (1,55,4,53)(2,18,3,25)(5,29,46,17)(6,19,47,26)(7,9,48,45)(8,34,49,44)(10,20,60,38)(11,57,61,50)(12,13,62,63)(14,58,64,51)(15,40,27,30)(16,43,28,33)(21,37,39,24)(22,59,35,52)(23,54,36,56)(31,32,41,42), (1,63)(2,42)(3,32)(4,13)(5,40)(6,16)(7,34)(8,9)(10,37)(11,22)(12,53)(14,56)(15,29)(17,27)(18,41)(19,33)(20,21)(23,51)(24,60)(25,31)(26,43)(28,47)(30,46)(35,61)(36,58)(38,39)(44,48)(45,49)(50,59)(52,57)(54,64)(55,62), (1,20,4,38)(2,49,3,8)(5,26,46,19)(6,17,47,29)(7,41,48,31)(9,42,45,32)(10,55,60,53)(11,14,61,64)(12,37,62,24)(13,39,63,21)(15,16,27,28)(18,44,25,34)(22,56,35,54)(23,52,36,59)(30,33,40,43)(50,51,57,58), (1,58,4,51)(2,16,3,28)(5,7,46,48)(6,32,47,42)(8,27,49,15)(9,17,45,29)(10,11,60,61)(12,56,62,54)(13,23,63,36)(14,55,64,53)(18,43,25,33)(19,41,26,31)(20,57,38,50)(21,52,39,59)(22,24,35,37)(30,44,40,34), (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( (1,3)(2,4)(5,22)(6,23)(7,24)(8,20)(9,21)(10,44)(11,40)(12,41)(13,42)(14,43)(15,50)(16,51)(17,52)(18,53)(19,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,61)(31,62)(32,63)(33,64)(34,60)(35,46)(36,47)(37,48)(38,49)(39,45), (1,55,4,53)(2,18,3,25)(5,29,46,17)(6,19,47,26)(7,9,48,45)(8,34,49,44)(10,20,60,38)(11,57,61,50)(12,13,62,63)(14,58,64,51)(15,40,27,30)(16,43,28,33)(21,37,39,24)(22,59,35,52)(23,54,36,56)(31,32,41,42), (1,63)(2,42)(3,32)(4,13)(5,40)(6,16)(7,34)(8,9)(10,37)(11,22)(12,53)(14,56)(15,29)(17,27)(18,41)(19,33)(20,21)(23,51)(24,60)(25,31)(26,43)(28,47)(30,46)(35,61)(36,58)(38,39)(44,48)(45,49)(50,59)(52,57)(54,64)(55,62), (1,20,4,38)(2,49,3,8)(5,26,46,19)(6,17,47,29)(7,41,48,31)(9,42,45,32)(10,55,60,53)(11,14,61,64)(12,37,62,24)(13,39,63,21)(15,16,27,28)(18,44,25,34)(22,56,35,54)(23,52,36,59)(30,33,40,43)(50,51,57,58), (1,58,4,51)(2,16,3,28)(5,7,46,48)(6,32,47,42)(8,27,49,15)(9,17,45,29)(10,11,60,61)(12,56,62,54)(13,23,63,36)(14,55,64,53)(18,43,25,33)(19,41,26,31)(20,57,38,50)(21,52,39,59)(22,24,35,37)(30,44,40,34), (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([[(1,3),(2,4),(5,22),(6,23),(7,24),(8,20),(9,21),(10,44),(11,40),(12,41),(13,42),(14,43),(15,50),(16,51),(17,52),(18,53),(19,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,61),(31,62),(32,63),(33,64),(34,60),(35,46),(36,47),(37,48),(38,49),(39,45)], [(1,55,4,53),(2,18,3,25),(5,29,46,17),(6,19,47,26),(7,9,48,45),(8,34,49,44),(10,20,60,38),(11,57,61,50),(12,13,62,63),(14,58,64,51),(15,40,27,30),(16,43,28,33),(21,37,39,24),(22,59,35,52),(23,54,36,56),(31,32,41,42)], [(1,63),(2,42),(3,32),(4,13),(5,40),(6,16),(7,34),(8,9),(10,37),(11,22),(12,53),(14,56),(15,29),(17,27),(18,41),(19,33),(20,21),(23,51),(24,60),(25,31),(26,43),(28,47),(30,46),(35,61),(36,58),(38,39),(44,48),(45,49),(50,59),(52,57),(54,64),(55,62)], [(1,20,4,38),(2,49,3,8),(5,26,46,19),(6,17,47,29),(7,41,48,31),(9,42,45,32),(10,55,60,53),(11,14,61,64),(12,37,62,24),(13,39,63,21),(15,16,27,28),(18,44,25,34),(22,56,35,54),(23,52,36,59),(30,33,40,43),(50,51,57,58)], [(1,58,4,51),(2,16,3,28),(5,7,46,48),(6,32,47,42),(8,27,49,15),(9,17,45,29),(10,11,60,61),(12,56,62,54),(13,23,63,36),(14,55,64,53),(18,43,25,33),(19,41,26,31),(20,57,38,50),(21,52,39,59),(22,24,35,37),(30,44,40,34)], [(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 of C2×2- 1+4⋊C5 ►in 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 |
1 | 0 | 0 | 0 | 0 |
0 | 39 | 5 | 24 | 38 |
0 | 5 | 39 | 38 | 24 |
0 | 24 | 38 | 2 | 36 |
0 | 38 | 24 | 36 | 2 |
1 | 0 | 0 | 0 | 0 |
0 | 2 | 20 | 7 | 29 |
0 | 21 | 39 | 12 | 34 |
0 | 7 | 29 | 39 | 21 |
0 | 12 | 34 | 20 | 2 |
1 | 0 | 0 | 0 | 0 |
0 | 20 | 36 | 13 | 30 |
0 | 5 | 21 | 11 | 28 |
0 | 28 | 11 | 20 | 36 |
0 | 30 | 13 | 5 | 21 |
1 | 0 | 0 | 0 | 0 |
0 | 12 | 34 | 20 | 2 |
0 | 7 | 29 | 39 | 21 |
0 | 20 | 2 | 29 | 7 |
0 | 39 | 21 | 34 | 12 |
37 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 7 | 29 | 39 | 21 |
0 | 0 | 0 | 0 | 40 |
0 | 21 | 39 | 12 | 34 |
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],[1,0,0,0,0,0,39,5,24,38,0,5,39,38,24,0,24,38,2,36,0,38,24,36,2],[1,0,0,0,0,0,2,21,7,12,0,20,39,29,34,0,7,12,39,20,0,29,34,21,2],[1,0,0,0,0,0,20,5,28,30,0,36,21,11,13,0,13,11,20,5,0,30,28,36,21],[1,0,0,0,0,0,12,7,20,39,0,34,29,2,21,0,20,39,29,34,0,2,21,7,12],[37,0,0,0,0,0,1,7,0,21,0,0,29,0,39,0,0,39,0,12,0,0,21,40,34] >;
C2×2- 1+4⋊C5 in GAP, Magma, Sage, TeX
C_2\times 2_-^{1+4}\rtimes C_5
% in TeX
G:=Group("C2xES-(2,2):C5");
// GroupNames label
G:=SmallGroup(320,1585);
// by ID
G=gap.SmallGroup(320,1585);
# by ID
G:=PCGroup([7,-2,-5,-2,2,2,2,-2,849,1270,521,248,1936,718,375,172,3162,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=f^5=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=b^2*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,e*d*e^-1=b^2*d,f*d*f^-1=b*c*d,f*e*f^-1=d*e>;
// generators/relations
Export