Copied to
clipboard

## G = C4.Dic5order 80 = 24·5

### The non-split extension by C4 of Dic5 acting via Dic5/C10=C2

Aliases: C4.Dic5, C20.4C4, C54M4(2), C4.15D10, C22.Dic5, C20.15C22, C52C85C2, (C2×C4).2D5, (C2×C20).5C2, (C2×C10).5C4, C10.14(C2×C4), C2.3(C2×Dic5), SmallGroup(80,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C4.Dic5
 Lower central C5 — C10 — C4.Dic5
 Upper central C1 — C4 — C2×C4

Generators and relations for C4.Dic5
G = < a,b,c | a4=1, b10=a2, c2=b5, ab=ba, cac-1=a-1, cbc-1=b9 >

Character table of C4.Dic5

 class 1 2A 2B 4A 4B 4C 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 20A 20B 20C 20D 20E 20F 20G 20H size 1 1 2 1 1 2 2 2 10 10 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 -1 -1 1 1 i i -i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 -1 1 1 1 -i i i -i 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 linear of order 4 ρ7 1 1 -1 -1 -1 1 1 1 i -i -i i 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 1 -i -i i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 2 2 -2 2 2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 1-√5/2 orthogonal lifted from D10 ρ10 2 2 2 2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 2 2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ12 2 2 -2 2 2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 1+√5/2 orthogonal lifted from D10 ρ13 2 2 2 -2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ14 2 2 -2 -2 -2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 -1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ15 2 2 -2 -2 -2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 -1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ16 2 2 2 -2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ17 2 -2 0 2i -2i 0 2 2 0 0 0 0 -2 0 0 0 0 -2 -2i 0 0 0 2i 2i -2i 0 complex lifted from M4(2) ρ18 2 -2 0 -2i 2i 0 2 2 0 0 0 0 -2 0 0 0 0 -2 2i 0 0 0 -2i -2i 2i 0 complex lifted from M4(2) ρ19 2 -2 0 -2i 2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ43ζ54-ζ43ζ5 complex faithful ρ20 2 -2 0 -2i 2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 ζ4ζ54+ζ4ζ5 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 complex faithful ρ21 2 -2 0 2i -2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 ζ43ζ53+ζ43ζ52 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 complex faithful ρ22 2 -2 0 -2i 2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 ζ4ζ54+ζ4ζ5 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 -ζ4ζ53+ζ4ζ52 complex faithful ρ23 2 -2 0 -2i 2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ4ζ53+ζ4ζ52 ζ43ζ54-ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 -ζ43ζ54+ζ43ζ5 complex faithful ρ24 2 -2 0 2i -2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 -ζ43ζ54+ζ43ζ5 -ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ4ζ53-ζ4ζ52 complex faithful ρ25 2 -2 0 2i -2i 0 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 ζ43ζ54+ζ43ζ5 -ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 ζ4ζ53-ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 -ζ4ζ53+ζ4ζ52 complex faithful ρ26 2 -2 0 2i -2i 0 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 ζ43ζ53+ζ43ζ52 -ζ4ζ53+ζ4ζ52 ζ4ζ53-ζ4ζ52 -ζ43ζ54+ζ43ζ5 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ54-ζ43ζ5 complex faithful

Smallest permutation representation of C4.Dic5
On 40 points
Generators in S40
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(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)
(1 26 6 31 11 36 16 21)(2 35 7 40 12 25 17 30)(3 24 8 29 13 34 18 39)(4 33 9 38 14 23 19 28)(5 22 10 27 15 32 20 37)

G:=sub<Sym(40)| (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (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), (1,26,6,31,11,36,16,21)(2,35,7,40,12,25,17,30)(3,24,8,29,13,34,18,39)(4,33,9,38,14,23,19,28)(5,22,10,27,15,32,20,37)>;

G:=Group( (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,36,31,26)(22,37,32,27)(23,38,33,28)(24,39,34,29)(25,40,35,30), (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), (1,26,6,31,11,36,16,21)(2,35,7,40,12,25,17,30)(3,24,8,29,13,34,18,39)(4,33,9,38,14,23,19,28)(5,22,10,27,15,32,20,37) );

G=PermutationGroup([(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,36,31,26),(22,37,32,27),(23,38,33,28),(24,39,34,29),(25,40,35,30)], [(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)], [(1,26,6,31,11,36,16,21),(2,35,7,40,12,25,17,30),(3,24,8,29,13,34,18,39),(4,33,9,38,14,23,19,28),(5,22,10,27,15,32,20,37)])

C4.Dic5 is a maximal subgroup of
D204C4  C40.6C4  C20.53D4  C20.46D4  C4.12D20  C20.D4  C20.10D4  D42Dic5  D20.3C4  D5×M4(2)  D4.D10  C20.C23  D4.Dic5  D4⋊D10  D4.9D10  D6.Dic5  C60.7C4  C4.Dic25  C20.30D10  C20.59D10  C20.12F5  C102.C4
C4.Dic5 is a maximal quotient of
C42.D5  C203C8  C20.55D4  D6.Dic5  C60.7C4  C4.Dic25  C20.30D10  C20.59D10  C20.12F5  C102.C4

Matrix representation of C4.Dic5 in GL2(𝔽41) generated by

 32 0 0 9
,
 36 0 0 33
,
 0 1 32 0
G:=sub<GL(2,GF(41))| [32,0,0,9],[36,0,0,33],[0,32,1,0] >;

C4.Dic5 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_5
% in TeX

G:=Group("C4.Dic5");
// GroupNames label

G:=SmallGroup(80,10);
// by ID

G=gap.SmallGroup(80,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,42,1604]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^10=a^2,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^9>;
// generators/relations

Export

׿
×
𝔽