p-group, metabelian, nilpotent (class 3), monomial
Aliases: C25:1C4, C24.64D4, C25.7C22, C24.164C23, (C23xC4):6C4, C24:3C4:1C2, (C2xD4).259D4, (C22xD4):15C4, (D4xC23).2C2, C22:2(C23:C4), C23:3(C22:C4), C24.112(C2xC4), C22.7C22wrC2, C23.543(C2xD4), C2.6(C24:3C4), C23.180(C22xC4), (C22xD4).448C22, (C2xC23:C4):1C2, (C2xC4):3(C22:C4), C2.24(C2xC23:C4), (C2xC22:C4):1C22, (C22xC4).72(C2xC4), C22.28(C2xC22:C4), SmallGroup(128,513)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C25:C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ace, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 1108 in 440 conjugacy classes, 72 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C22:C4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C24, C24, C23:C4, C2xC22:C4, C2xC22:C4, C23xC4, C22xD4, C22xD4, C25, C24:3C4, C2xC23:C4, D4xC23, C25:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C23:C4, C2xC22:C4, C22wrC2, C24:3C4, C2xC23:C4, C25:C4
(1 7)(3 12)(4 14)(5 13)(6 9)(10 15)
(1 15)(2 11)(3 13)(4 9)(5 12)(6 14)(7 10)(8 16)
(1 10)(2 8)(3 12)(4 6)(5 13)(7 15)(9 14)(11 16)
(1 7)(2 8)(3 5)(4 6)(9 14)(10 15)(11 16)(12 13)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,7)(3,12)(4,14)(5,13)(6,9)(10,15), (1,15)(2,11)(3,13)(4,9)(5,12)(6,14)(7,10)(8,16), (1,10)(2,8)(3,12)(4,6)(5,13)(7,15)(9,14)(11,16), (1,7)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,7)(3,12)(4,14)(5,13)(6,9)(10,15), (1,15)(2,11)(3,13)(4,9)(5,12)(6,14)(7,10)(8,16), (1,10)(2,8)(3,12)(4,6)(5,13)(7,15)(9,14)(11,16), (1,7)(2,8)(3,5)(4,6)(9,14)(10,15)(11,16)(12,13), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,7),(3,12),(4,14),(5,13),(6,9),(10,15)], [(1,15),(2,11),(3,13),(4,9),(5,12),(6,14),(7,10),(8,16)], [(1,10),(2,8),(3,12),(4,6),(5,13),(7,15),(9,14),(11,16)], [(1,7),(2,8),(3,5),(4,6),(9,14),(10,15),(11,16),(12,13)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,240);
(1 14)(2 15)(3 12)(4 11)(5 16)(6 13)(7 10)(8 9)
(1 8)(4 5)(9 14)(11 16)
(1 5)(4 8)(9 11)(14 16)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 15)(11 16)(12 13)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,14)(2,15)(3,12)(4,11)(5,16)(6,13)(7,10)(8,9), (1,8)(4,5)(9,14)(11,16), (1,5)(4,8)(9,11)(14,16), (1,8)(2,7)(3,6)(4,5)(9,14)(10,15)(11,16)(12,13), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,14)(2,15)(3,12)(4,11)(5,16)(6,13)(7,10)(8,9), (1,8)(4,5)(9,14)(11,16), (1,5)(4,8)(9,11)(14,16), (1,8)(2,7)(3,6)(4,5)(9,14)(10,15)(11,16)(12,13), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([[(1,14),(2,15),(3,12),(4,11),(5,16),(6,13),(7,10),(8,9)], [(1,8),(4,5),(9,14),(11,16)], [(1,5),(4,8),(9,11),(14,16)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,15),(11,16),(12,13)], [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)]])
G:=TransitiveGroup(16,275);
32 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2S | 4A | 4B | 4C | 4D | 4E | ··· | 4L |
order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | C23:C4 |
kernel | C25:C4 | C24:3C4 | C2xC23:C4 | D4xC23 | C23xC4 | C22xD4 | C25 | C2xD4 | C24 | C22 |
# reps | 1 | 2 | 4 | 1 | 2 | 4 | 2 | 8 | 4 | 4 |
Matrix representation of C25:C4 ►in GL6(F5)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
4 | 0 | 0 | 0 | 0 | 0 |
0 | 4 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 4 | 0 | 0 | 0 |
0 | 0 | 0 | 4 | 0 | 0 |
0 | 0 | 0 | 0 | 4 | 0 |
0 | 0 | 0 | 0 | 0 | 4 |
3 | 0 | 0 | 0 | 0 | 0 |
0 | 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 4 | 3 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C25:C4 in GAP, Magma, Sage, TeX
C_2^5\rtimes C_4
% in TeX
G:=Group("C2^5:C4");
// GroupNames label
G:=SmallGroup(128,513);
// by ID
G=gap.SmallGroup(128,513);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,2028]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f^-1=a*c*e,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations