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G = C4⋊Dic5  order 80 = 24·5

The semidirect product of C4 and Dic5 acting via Dic5/C10=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic5, C20⋊3C4, C2.1D20, C10.4D4, C10.2Q8, C2.2Dic10, C22.5D10, C5⋊3(C4⋊C4), (C2×C4).3D5, (C2×C20).3C2, C10.15(C2×C4), C2.4(C2×Dic5), (C2×C10).5C22, (C2×Dic5).2C2, SmallGroup(80,13)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C4⋊Dic5
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4⋊Dic5
Lower central
C5 — C10 — C4⋊Dic5
Upper central
C1 — C22 — C2×C4

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

C1
C2
C2
C2
C5
C4
C4
C22
10C4
10C4
C10
C10
C10
C2×C4
5C2×C4
5C2×C4
C2×C10
C20
C20
2Dic5
2Dic5
5C4⋊C4
C2×Dic5
C2×C20
C2×Dic5
C4⋊Dic5

Character table of C4⋊Dic5

 class 12A2B2C4A4B4C4D4E4F5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 111122101010102222222222222222
ρ111111111111111111111111111    trivial
ρ21111-1-1-11-1111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11-11-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-11111111111111111    linear of order 2
ρ51-1-11-11i-i-ii11-11-1-1-111111-1-1-1-1    linear of order 4
ρ61-1-111-1-i-iii11-11-1-1-11-1-1-1-11111    linear of order 4
ρ71-1-111-1ii-i-i11-11-1-1-11-1-1-1-11111    linear of order 4
ρ81-1-11-11-iii-i11-11-1-1-111111-1-1-1-1    linear of order 4
ρ92-22-200000022-2-2-222-200000000    orthogonal lifted from D4
ρ102222-2-20000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ112222-2-20000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ122222220000-1+√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-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ132222220000-1-√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+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ142-22-2000000-1-√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/21+√5/2-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    orthogonal lifted from D20
ρ152-22-2000000-1+√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/21-√5/2ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    orthogonal lifted from D20
ρ162-22-2000000-1+√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/21-√5/2-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    orthogonal lifted from D20
ρ172-22-2000000-1-√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/21+√5/2ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    orthogonal lifted from D20
ρ1822-2-2000000222-22-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ192-2-22-220000-1-√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/21-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21-√5/21-√5/21+√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ202-2-222-20000-1-√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/21-√5/2-1-√5/21+√5/21-√5/21-√5/21+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    symplectic lifted from Dic5, Schur index 2
ρ212-2-222-20000-1+√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/21+√5/2-1+√5/21-√5/21+√5/21+√5/21-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    symplectic lifted from Dic5, Schur index 2
ρ222-2-22-220000-1+√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/21+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21+√5/21+√5/21-√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ2322-2-2000000-1-√5/2-1+√5/2-1-√5/21-√5/2-1+√5/21+√5/21-√5/21+√5/2ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52    symplectic lifted from Dic10, Schur index 2
ρ2422-2-2000000-1+√5/2-1-√5/2-1+√5/21+√5/2-1-√5/21-√5/21+√5/21-√5/2ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5    symplectic lifted from Dic10, Schur index 2
ρ2522-2-2000000-1+√5/2-1-√5/2-1+√5/21+√5/2-1-√5/21-√5/21+√5/21-√5/2-ζ43ζ54+ζ43ζ5ζ4ζ53-ζ4ζ52-ζ4ζ53+ζ4ζ52ζ43ζ54-ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5    symplectic lifted from Dic10, Schur index 2
ρ2622-2-2000000-1-√5/2-1+√5/2-1-√5/21-√5/2-1+√5/21+√5/21-√5/21+√5/2-ζ4ζ53+ζ4ζ52-ζ43ζ54+ζ43ζ5ζ43ζ54-ζ43ζ5ζ4ζ53-ζ4ζ52ζ43ζ54-ζ43ζ5-ζ43ζ54+ζ43ζ5-ζ4ζ53+ζ4ζ52ζ4ζ53-ζ4ζ52    symplectic lifted from Dic10, Schur index 2

Smallest permutation representation of C4⋊Dic5
►Regular action on 80 points
Generators in S80
(1 45 28 34)(2 46 29 35)(3 47 30 36)(4 48 21 37)(5 49 22 38)(6 50 23 39)(7 41 24 40)(8 42 25 31)(9 43 26 32)(10 44 27 33)(11 51 71 62)(12 52 72 63)(13 53 73 64)(14 54 74 65)(15 55 75 66)(16 56 76 67)(17 57 77 68)(18 58 78 69)(19 59 79 70)(20 60 80 61)

(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 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)

(1 51 6 56)(2 60 7 55)(3 59 8 54)(4 58 9 53)(5 57 10 52)(11 50 16 45)(12 49 17 44)(13 48 18 43)(14 47 19 42)(15 46 20 41)(21 69 26 64)(22 68 27 63)(23 67 28 62)(24 66 29 61)(25 65 30 70)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
 
G:=sub<Sym(80)| (1,45,28,34)(2,46,29,35)(3,47,30,36)(4,48,21,37)(5,49,22,38)(6,50,23,39)(7,41,24,40)(8,42,25,31)(9,43,26,32)(10,44,27,33)(11,51,71,62)(12,52,72,63)(13,53,73,64)(14,54,74,65)(15,55,75,66)(16,56,76,67)(17,57,77,68)(18,58,78,69)(19,59,79,70)(20,60,80,61), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,51,6,56)(2,60,7,55)(3,59,8,54)(4,58,9,53)(5,57,10,52)(11,50,16,45)(12,49,17,44)(13,48,18,43)(14,47,19,42)(15,46,20,41)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75)>;
 
G:=Group( (1,45,28,34)(2,46,29,35)(3,47,30,36)(4,48,21,37)(5,49,22,38)(6,50,23,39)(7,41,24,40)(8,42,25,31)(9,43,26,32)(10,44,27,33)(11,51,71,62)(12,52,72,63)(13,53,73,64)(14,54,74,65)(15,55,75,66)(16,56,76,67)(17,57,77,68)(18,58,78,69)(19,59,79,70)(20,60,80,61), (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)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80), (1,51,6,56)(2,60,7,55)(3,59,8,54)(4,58,9,53)(5,57,10,52)(11,50,16,45)(12,49,17,44)(13,48,18,43)(14,47,19,42)(15,46,20,41)(21,69,26,64)(22,68,27,63)(23,67,28,62)(24,66,29,61)(25,65,30,70)(31,74,36,79)(32,73,37,78)(33,72,38,77)(34,71,39,76)(35,80,40,75) );
 
G=PermutationGroup([[(1,45,28,34),(2,46,29,35),(3,47,30,36),(4,48,21,37),(5,49,22,38),(6,50,23,39),(7,41,24,40),(8,42,25,31),(9,43,26,32),(10,44,27,33),(11,51,71,62),(12,52,72,63),(13,53,73,64),(14,54,74,65),(15,55,75,66),(16,56,76,67),(17,57,77,68),(18,58,78,69),(19,59,79,70),(20,60,80,61)], [(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),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80)], [(1,51,6,56),(2,60,7,55),(3,59,8,54),(4,58,9,53),(5,57,10,52),(11,50,16,45),(12,49,17,44),(13,48,18,43),(14,47,19,42),(15,46,20,41),(21,69,26,64),(22,68,27,63),(23,67,28,62),(24,66,29,61),(25,65,30,70),(31,74,36,79),(32,73,37,78),(33,72,38,77),(34,71,39,76),(35,80,40,75)]])
 

C4⋊Dic5 is a maximal subgroup of
 C10.D8  C20.Q8  C20.44D4  C40⋊6C4  C40⋊5C4  D20⋊5C4  D4⋊Dic5  Q8⋊Dic5  C4×Dic10  C20⋊2Q8  C20.6Q8  C4×D20  Dic5.14D4  C23.D10  D10.12D4  C22.D20  C20⋊Q8  Dic5.Q8  C4.Dic10  D5×C4⋊C4  C4⋊C4⋊7D5  D10⋊2Q8  C4⋊C4⋊D5  C20.48D4  C23.21D10  C20⋊7D4  D4×Dic5  C20⋊2D4  Q8×Dic5  D10⋊3Q8  C6.Dic10  C60⋊5C4  C4⋊Dic25  Dic5⋊Dic5  C20⋊3Dic5  C20⋊5F5
C4⋊Dic5 is a maximal quotient of
 C20⋊3C8  C40⋊6C4  C40⋊5C4  C40.6C4  C10.10C42  C6.Dic10  C60⋊5C4  C4⋊Dic25  Dic5⋊Dic5  C20⋊3Dic5  C20⋊5F5

Matrix representation of C4⋊Dic5 ►in GL3(𝔽41) generated by

4000
03932
0372
,
4000
0140
0366
,
900
0638
02635
G:=sub<GL(3,GF(41))| [40,0,0,0,39,37,0,32,2],[40,0,0,0,1,36,0,40,6],[9,0,0,0,6,26,0,38,35] >;
 

C4⋊Dic5 in GAP, Magma, Sage, TeX 

C_4\rtimes {\rm Dic}_5
 
% in TeX
 
G:=Group("C4:Dic5");
 
// GroupNames label
 
G:=SmallGroup(80,13);
 
// by ID
 
G=gap.SmallGroup(80,13);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,46,1604]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^10=1,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4⋊Dic5 in TeX
Character table of C4⋊Dic5 in TeX

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