direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C25⋊C4, C50⋊C4, D25⋊C4, D50.C2, C10.2F5, D25.C22, C25⋊(C2×C4), C5.(C2×F5), SmallGroup(200,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C2×C25⋊C4 |
Generators and relations for C2×C25⋊C4
G = < a,b,c | a2=b25=c4=1, ab=ba, ac=ca, cbc-1=b18 >
Character table of C2×C25⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 10 | 25A | 25B | 25C | 25D | 25E | 50A | 50B | 50C | 50D | 50E | |
| size | 1 | 1 | 25 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ10 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | orthogonal lifted from C25⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | -ζ2524-ζ2518-ζ257-ζ25 | -ζ2519-ζ2517-ζ258-ζ256 | -ζ2523-ζ2514-ζ2511-ζ252 | -ζ2516-ζ2513-ζ2512-ζ259 | -ζ2522-ζ2521-ζ254-ζ253 | orthogonal faithful |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | orthogonal lifted from C25⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | orthogonal lifted from C25⋊C4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | -ζ2522-ζ2521-ζ254-ζ253 | -ζ2524-ζ2518-ζ257-ζ25 | -ζ2519-ζ2517-ζ258-ζ256 | -ζ2523-ζ2514-ζ2511-ζ252 | -ζ2516-ζ2513-ζ2512-ζ259 | orthogonal faithful |
| ρ16 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | orthogonal lifted from C25⋊C4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | -ζ2519-ζ2517-ζ258-ζ256 | -ζ2523-ζ2514-ζ2511-ζ252 | -ζ2516-ζ2513-ζ2512-ζ259 | -ζ2522-ζ2521-ζ254-ζ253 | -ζ2524-ζ2518-ζ257-ζ25 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | -ζ2516-ζ2513-ζ2512-ζ259 | -ζ2522-ζ2521-ζ254-ζ253 | -ζ2524-ζ2518-ζ257-ζ25 | -ζ2519-ζ2517-ζ258-ζ256 | -ζ2523-ζ2514-ζ2511-ζ252 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | -ζ2523-ζ2514-ζ2511-ζ252 | -ζ2516-ζ2513-ζ2512-ζ259 | -ζ2522-ζ2521-ζ254-ζ253 | -ζ2524-ζ2518-ζ257-ζ25 | -ζ2519-ζ2517-ζ258-ζ256 | orthogonal faithful |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | orthogonal lifted from C25⋊C4 |
►On 50 points
Generators in S50
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)
(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)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 48 27 30)(28 37 50 41)(29 44 49 34)(31 33 47 45)(32 40 46 38)(35 36 43 42)
G:=sub<Sym(50)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38), (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), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,48,27,30)(28,37,50,41)(29,44,49,34)(31,33,47,45)(32,40,46,38)(35,36,43,42)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,35)(23,36)(24,37)(25,38), (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), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,48,27,30)(28,37,50,41)(29,44,49,34)(31,33,47,45)(32,40,46,38)(35,36,43,42) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,35),(23,36),(24,37),(25,38)], [(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)], [(2,8,25,19),(3,15,24,12),(4,22,23,5),(6,11,21,16),(7,18,20,9),(10,14,17,13),(26,48,27,30),(28,37,50,41),(29,44,49,34),(31,33,47,45),(32,40,46,38),(35,36,43,42)]])
C2×C25⋊C4 is a maximal subgroup of
C100⋊C4 D25.D4
C2×C25⋊C4 is a maximal quotient of D25⋊C8 C100.C4 C100⋊C4 C25⋊M4(2) D25.D4
Matrix representation of C2×C25⋊C4 ►in GL4(𝔽7) generated by
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 4 | 2 | 1 | 2 |
| 3 | 0 | 4 | 0 |
| 0 | 5 | 0 | 0 |
| 6 | 0 | 5 | 6 |
| 1 | 3 | 2 | 3 |
| 4 | 3 | 2 | 2 |
| 0 | 2 | 2 | 0 |
| 0 | 4 | 2 | 1 |
G:=sub<GL(4,GF(7))| [6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[4,3,0,6,2,0,5,0,1,4,0,5,2,0,0,6],[1,4,0,0,3,3,2,4,2,2,2,2,3,2,0,1] >;
C2×C25⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{25}\rtimes C_4 % in TeX
G:=Group("C2xC25:C4"); // GroupNames label
G:=SmallGroup(200,12);
// by ID
G=gap.SmallGroup(200,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,1123,973,118,2004,1014]);
// Polycyclic
G:=Group<a,b,c|a^2=b^25=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations
Export
Subgroup lattice of C2×C25⋊C4 in TeX
Character table of C2×C25⋊C4 in TeX