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G = C4⋊Dic5order 80 = 24·5

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

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊Dic5, C203C4, C2.1D20, C10.4D4, C10.2Q8, C2.2Dic10, C22.5D10, C53(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

C1C10 — C4⋊Dic5
C1C5C10C2×C10C2×Dic5 — C4⋊Dic5
C5C10 — C4⋊Dic5
C1C22C2×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 >

10C4
10C4
5C2×C4
5C2×C4
2Dic5
2Dic5
5C4⋊C4

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+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    orthogonal lifted from D20
ρ152-22-2000000-1+5/2-1-5/21-5/21+5/21+5/2-1+5/2-1-5/21-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    orthogonal lifted from D20
ρ162-22-2000000-1+5/2-1-5/21-5/21+5/21+5/2-1+5/2-1-5/21-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    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+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    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+5/2ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    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+5/21-5/2ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    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+5/21-5/243ζ5443ζ5ζ4ζ534ζ524ζ534ζ52ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    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+5/24ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    symplectic lifted from Dic10, Schur index 2

Smallest permutation representation of C4⋊Dic5
Regular action on 80 points
Generators in S80
(1 45 28 36)(2 46 29 37)(3 47 30 38)(4 48 21 39)(5 49 22 40)(6 50 23 31)(7 41 24 32)(8 42 25 33)(9 43 26 34)(10 44 27 35)(11 59 79 62)(12 60 80 63)(13 51 71 64)(14 52 72 65)(15 53 73 66)(16 54 74 67)(17 55 75 68)(18 56 76 69)(19 57 77 70)(20 58 78 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 59 6 54)(2 58 7 53)(3 57 8 52)(4 56 9 51)(5 55 10 60)(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)| (1,45,28,36)(2,46,29,37)(3,47,30,38)(4,48,21,39)(5,49,22,40)(6,50,23,31)(7,41,24,32)(8,42,25,33)(9,43,26,34)(10,44,27,35)(11,59,79,62)(12,60,80,63)(13,51,71,64)(14,52,72,65)(15,53,73,66)(16,54,74,67)(17,55,75,68)(18,56,76,69)(19,57,77,70)(20,58,78,61), (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), (1,59,6,54)(2,58,7,53)(3,57,8,52)(4,56,9,51)(5,55,10,60)(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( (1,45,28,36)(2,46,29,37)(3,47,30,38)(4,48,21,39)(5,49,22,40)(6,50,23,31)(7,41,24,32)(8,42,25,33)(9,43,26,34)(10,44,27,35)(11,59,79,62)(12,60,80,63)(13,51,71,64)(14,52,72,65)(15,53,73,66)(16,54,74,67)(17,55,75,68)(18,56,76,69)(19,57,77,70)(20,58,78,61), (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), (1,59,6,54)(2,58,7,53)(3,57,8,52)(4,56,9,51)(5,55,10,60)(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([(1,45,28,36),(2,46,29,37),(3,47,30,38),(4,48,21,39),(5,49,22,40),(6,50,23,31),(7,41,24,32),(8,42,25,33),(9,43,26,34),(10,44,27,35),(11,59,79,62),(12,60,80,63),(13,51,71,64),(14,52,72,65),(15,53,73,66),(16,54,74,67),(17,55,75,68),(18,56,76,69),(19,57,77,70),(20,58,78,61)], [(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)], [(1,59,6,54),(2,58,7,53),(3,57,8,52),(4,56,9,51),(5,55,10,60),(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  C406C4  C405C4  D205C4  D4⋊Dic5  Q8⋊Dic5  C4×Dic10  C202Q8  C20.6Q8  C4×D20  Dic5.14D4  C23.D10  D10.12D4  C22.D20  C20⋊Q8  Dic5.Q8  C4.Dic10  D5×C4⋊C4  C4⋊C47D5  D102Q8  C4⋊C4⋊D5  C20.48D4  C23.21D10  C207D4  D4×Dic5  C202D4  Q8×Dic5  D103Q8  C6.Dic10  C605C4  C4⋊Dic25  Dic5⋊Dic5  C203Dic5  C205F5
C4⋊Dic5 is a maximal quotient of
C203C8  C406C4  C405C4  C40.6C4  C10.10C42  C6.Dic10  C605C4  C4⋊Dic25  Dic5⋊Dic5  C203Dic5  C205F5

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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