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G = D10⋊C4  order 80 = 24·5

1st semidirect product of D10 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10⋊1C4, C10.6D4, C2.2D20, C22.6D10, (C2×C4)⋊1D5, (C2×C20)⋊1C2, C2.5(C4×D5), C5⋊2(C22⋊C4), C10.12(C2×C4), (C2×Dic5)⋊1C2, C2.2(C5⋊D4), (C2×C10).6C22, (C22×D5).1C2, SmallGroup(80,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D10⋊C4
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — D10⋊C4
Lower central
C5 — C10 — D10⋊C4
Upper central
C1 — C22 — C2×C4

Generators and relations for D10⋊C4
 G = < a,b,c | a10=b2=c4=1, bab=a-1, ac=ca, cbc-1=a5b >

C1
C2
C2
C2
10C2
10C2
C5
C22
2C4
5C22
5C22
10C4
10C22
10C22
C10
C10
C10
2D5
2D5
C2×C4
5C2×C4
5C23
C2×C10
D10
D10
2D10
2D10
2Dic5
2C20
5C22⋊C4
C2×Dic5
C2×C20
C22×D5
D10⋊C4

Character table of D10⋊C4

 class 12A2B2C2D2E4A4B4C4D5A5B10A10B10C10D10E10F20A20B20C20D20E20F20G20H
 size 111110102210102222222222222222
ρ111111111111111111111111111    trivial
ρ21111-1-111-1-11111111111111111    linear of order 2
ρ31111-1-1-1-11111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ4111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-1-11-iii-i11-11-1-1-11iiii-i-i-i-i    linear of order 4
ρ611-1-11-1-ii-ii11-11-1-1-11iiii-i-i-i-i    linear of order 4
ρ711-1-11-1i-ii-i11-11-1-1-11-i-i-i-iiiii    linear of order 4
ρ811-1-1-11i-i-ii11-11-1-1-11-i-i-i-iiiii    linear of order 4
ρ92-22-200000022-2-2-222-200000000    orthogonal lifted from D4
ρ102-2-22000000222-22-2-2-200000000    orthogonal lifted from D4
ρ11222200-2-200-1-√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
ρ122-2-22000000-1+√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    orthogonal lifted from D20
ρ13222200-2-200-1+√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
ρ142222002200-1-√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
ρ152222002200-1+√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
ρ162-2-22000000-1-√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    orthogonal lifted from D20
ρ172-2-22000000-1-√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    orthogonal lifted from D20
ρ182-2-22000000-1+√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    orthogonal lifted from D20
ρ192-22-2000000-1+√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/21-√5/2ζ54-ζ5ζ53-ζ52-ζ53+ζ52-ζ54+ζ5ζ53-ζ52-ζ53+ζ52-ζ54+ζ5ζ54-ζ5    complex lifted from C5⋊D4
ρ202-22-2000000-1-√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/21+√5/2ζ53-ζ52-ζ54+ζ5ζ54-ζ5-ζ53+ζ52-ζ54+ζ5ζ54-ζ5-ζ53+ζ52ζ53-ζ52    complex lifted from C5⋊D4
ρ2122-2-200-2i2i00-1-√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/21-√5/2-1-√5/2ζ4ζ53+ζ4ζ52ζ4ζ54+ζ4ζ5ζ4ζ54+ζ4ζ5ζ4ζ53+ζ4ζ52ζ43ζ54+ζ43ζ5ζ43ζ54+ζ43ζ5ζ43ζ53+ζ43ζ52ζ43ζ53+ζ43ζ52    complex lifted from C4×D5
ρ222-22-2000000-1+√5/2-1-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/21-√5/2-ζ54+ζ5-ζ53+ζ52ζ53-ζ52ζ54-ζ5-ζ53+ζ52ζ53-ζ52ζ54-ζ5-ζ54+ζ5    complex lifted from C5⋊D4
ρ2322-2-2002i-2i00-1-√5/2-1+√5/21+√5/2-1+√5/21-√5/21+√5/21-√5/2-1-√5/2ζ43ζ53+ζ43ζ52ζ43ζ54+ζ43ζ5ζ43ζ54+ζ43ζ5ζ43ζ53+ζ43ζ52ζ4ζ54+ζ4ζ5ζ4ζ54+ζ4ζ5ζ4ζ53+ζ4ζ52ζ4ζ53+ζ4ζ52    complex lifted from C4×D5
ρ242-22-2000000-1-√5/2-1+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/21+√5/2-ζ53+ζ52ζ54-ζ5-ζ54+ζ5ζ53-ζ52ζ54-ζ5-ζ54+ζ5ζ53-ζ52-ζ53+ζ52    complex lifted from C5⋊D4
ρ2522-2-200-2i2i00-1+√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/21+√5/2-1+√5/2ζ4ζ54+ζ4ζ5ζ4ζ53+ζ4ζ52ζ4ζ53+ζ4ζ52ζ4ζ54+ζ4ζ5ζ43ζ53+ζ43ζ52ζ43ζ53+ζ43ζ52ζ43ζ54+ζ43ζ5ζ43ζ54+ζ43ζ5    complex lifted from C4×D5
ρ2622-2-2002i-2i00-1+√5/2-1-√5/21-√5/2-1-√5/21+√5/21-√5/21+√5/2-1+√5/2ζ43ζ54+ζ43ζ5ζ43ζ53+ζ43ζ52ζ43ζ53+ζ43ζ52ζ43ζ54+ζ43ζ5ζ4ζ53+ζ4ζ52ζ4ζ53+ζ4ζ52ζ4ζ54+ζ4ζ5ζ4ζ54+ζ4ζ5    complex lifted from C4×D5

Smallest permutation representation of D10⋊C4
►On 40 points
Generators in S40
(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 12)(2 11)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 13)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(30 40)

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

D10⋊C4 is a maximal subgroup of
 C42⋊D5  C4×D20  C4.D20  C42⋊2D5  D5×C22⋊C4  Dic5⋊4D4  C22⋊D20  D10.12D4  D10⋊D4  Dic5.5D4  C22.D20  C4⋊C4⋊7D5  D20⋊8C4  D10.13D4  C4⋊D20  D10⋊Q8  D10⋊2Q8  C4⋊C4⋊D5  C4×C5⋊D4  C23.23D10  C20⋊7D4  C23⋊D10  Dic5⋊D4  D10⋊3Q8  C20.23D4  D10⋊Dic3  D30⋊4C4  D30⋊3C4  D50⋊C4  D10⋊Dic5  C10.D20  C10.11D20  D5.D20
D10⋊C4 is a maximal quotient of
 D20⋊4C4  C23.1D10  D20⋊6C4  C10.Q16  C20.44D4  D10⋊1C8  D20⋊5C4  C20.46D4  C4.12D20  D20⋊7C4  C10.10C42  D10⋊Dic3  D30⋊4C4  D30⋊3C4  D50⋊C4  D10⋊Dic5  C10.D20  C10.11D20  D5.D20

Matrix representation of D10⋊C4 ►in GL3(𝔽41) generated by

100
03434
071
,
100
03434
017
,
900
01740
0124
G:=sub<GL(3,GF(41))| [1,0,0,0,34,7,0,34,1],[1,0,0,0,34,1,0,34,7],[9,0,0,0,17,1,0,40,24] >;
 

D10⋊C4 in GAP, Magma, Sage, TeX 

D_{10}\rtimes C_4
 
% in TeX
 
G:=Group("D10:C4");
 
// GroupNames label
 
G:=SmallGroup(80,14);
 
// by ID
 
G=gap.SmallGroup(80,14);
 
# by ID
 
G:=PCGroup([5,-2,-2,-2,-2,-5,101,26,1604]);
 
// Polycyclic
 
G:=Group<a,b,c|a^10=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^5*b>;
 
// generators/relations
 

Export

Subgroup lattice of D10⋊C4 in TeX
Character table of D10⋊C4 in TeX

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