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G = Dic20⋊C4order 320 = 26·5

3rd semidirect product of Dic20 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic20⋊C4
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4×F5 — Q8×F5 — Dic20⋊C4
 Lower central C5 — C10 — C20 — Dic20⋊C4
 Upper central C1 — C2 — C4 — Q16

Generators and relations for Dic20⋊C4
G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a13, cbc-1=a20b >

Subgroups: 418 in 108 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×2], C4, C4 [×9], C22, C5, C8, C8 [×2], C2×C4 [×7], Q8 [×2], Q8 [×4], D5 [×2], C10, C42 [×3], C4⋊C4 [×4], C2×C8 [×2], Q16, Q16 [×3], C2×Q8 [×2], Dic5, Dic5 [×2], C20, C20 [×2], F5 [×4], D10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8 [×2], C2×Q16, C52C8, C40, C5⋊C8, Dic10 [×2], Dic10 [×2], C4×D5, C4×D5 [×2], C5×Q8 [×2], C2×F5 [×2], C2×F5 [×2], Q16⋊C4, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, D5⋊C8, C4×F5, C4×F5 [×2], C4⋊F5 [×2], C4⋊F5 [×2], Q8×D5 [×2], C8⋊F5, C40⋊C4, Q8⋊F5 [×2], D5×Q16, Q8×F5 [×2], Dic20⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C8.C22 [×2], C2×F5 [×3], Q16⋊C4, C22×F5, D4×F5, Dic20⋊C4

Character table of Dic20⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5 8A 8B 8C 8D 10 20A 20B 20C 40A 40B size 1 1 5 5 2 4 4 10 10 10 10 10 20 20 20 20 20 20 4 4 20 20 20 4 8 16 16 8 8 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 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 -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 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 -1 1 1 linear of order 2 ρ5 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 -1 -1 -1 linear of order 2 ρ6 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 1 1 1 linear of order 2 ρ7 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 -1 1 1 linear of order 2 ρ8 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 1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 1 -1 1 -i -1 i i -i -1 1 -i i i -i 1 -1 1 -i i 1 1 1 -1 -1 -1 linear of order 4 ρ10 1 1 -1 -1 1 -1 -1 i -1 -i -i i 1 1 i -i i -i 1 1 -1 -i i 1 1 -1 -1 1 1 linear of order 4 ρ11 1 1 -1 -1 1 1 -1 -i -1 i i -i 1 -1 i -i -i i 1 -1 1 -i i 1 1 -1 1 -1 -1 linear of order 4 ρ12 1 1 -1 -1 1 1 1 i -1 -i -i i -1 -1 -i i -i i 1 1 -1 -i i 1 1 1 1 1 1 linear of order 4 ρ13 1 1 -1 -1 1 1 -1 i -1 -i -i i 1 -1 -i i i -i 1 -1 1 i -i 1 1 -1 1 -1 -1 linear of order 4 ρ14 1 1 -1 -1 1 1 1 -i -1 i i -i -1 -1 i -i i -i 1 1 -1 i -i 1 1 1 1 1 1 linear of order 4 ρ15 1 1 -1 -1 1 -1 1 i -1 -i -i i -1 1 i -i -i i 1 -1 1 i -i 1 1 1 -1 -1 -1 linear of order 4 ρ16 1 1 -1 -1 1 -1 -1 -i -1 i i -i 1 1 -i i -i i 1 1 -1 i -i 1 1 -1 -1 1 1 linear of order 4 ρ17 2 2 2 2 -2 0 0 -2 -2 2 -2 2 0 0 0 0 0 0 2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 0 0 2 -2 -2 2 -2 0 0 0 0 0 0 2 0 0 0 0 2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 -2 -2 -2 0 0 -2i 2 -2i 2i 2i 0 0 0 0 0 0 2 0 0 0 0 2 -2 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 -2 -2 -2 0 0 2i 2 2i -2i -2i 0 0 0 0 0 0 2 0 0 0 0 2 -2 0 0 0 0 complex lifted from C4○D4 ρ21 4 4 0 0 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 -4 0 0 0 -1 -1 1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 -1 -1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ23 4 4 0 0 4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 -4 0 0 0 -1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ24 4 4 0 0 4 4 4 0 0 0 0 0 0 0 0 0 0 0 -1 4 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 -4 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ27 8 8 0 0 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 -2 2 0 0 0 0 orthogonal lifted from D4×F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 -√10 √10 symplectic faithful, Schur index 2 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 2 0 0 0 √10 -√10 symplectic faithful, Schur index 2

Smallest permutation representation of Dic20⋊C4
On 80 points
Generators in S80
(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 50 21 70)(2 49 22 69)(3 48 23 68)(4 47 24 67)(5 46 25 66)(6 45 26 65)(7 44 27 64)(8 43 28 63)(9 42 29 62)(10 41 30 61)(11 80 31 60)(12 79 32 59)(13 78 33 58)(14 77 34 57)(15 76 35 56)(16 75 36 55)(17 74 37 54)(18 73 38 53)(19 72 39 52)(20 71 40 51)
(1 75 21 55)(2 72 30 68)(3 69 39 41)(4 66 8 54)(5 63 17 67)(6 60 26 80)(7 57 35 53)(9 51 13 79)(10 48 22 52)(11 45 31 65)(12 42 40 78)(14 76 18 64)(15 73 27 77)(16 70 36 50)(19 61 23 49)(20 58 32 62)(24 46 28 74)(25 43 37 47)(29 71 33 59)(34 56 38 44)

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

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

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

Matrix representation of Dic20⋊C4 in GL8(𝔽41)

 40 1 0 0 0 0 0 0 40 0 1 0 0 0 0 0 40 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 2 39 27 0 0 0 0 0 29 29 27 13 0 0 0 0 2 16 0 4 0 0 0 0 37 39 12 10
,
 0 0 0 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 0 0 0 0 0 0 0 40 0 0 0 0 0 1 1 1 2 0 0 0 0 1 0 0 0 0 0 0 0 40 40 0 40
,
 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 25 39 37 37 0 0 0 0 39 25 0 37 0 0 0 0 39 2 14 0 0 0 0 0 16 14 2 18

G:=sub<GL(8,GF(41))| [40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,29,2,37,0,0,0,0,39,29,16,39,0,0,0,0,27,27,0,12,0,0,0,0,0,13,4,10],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,1,40,0,0,0,0,0,1,0,40,0,0,0,0,40,1,0,0,0,0,0,0,0,2,0,40],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,25,39,39,16,0,0,0,0,39,25,2,14,0,0,0,0,37,0,14,2,0,0,0,0,37,37,0,18] >;

Dic20⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}\rtimes C_4
% in TeX

G:=Group("Dic20:C4");
// GroupNames label

G:=SmallGroup(320,1077);
// by ID

G=gap.SmallGroup(320,1077);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^40=c^4=1,b^2=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^20*b>;
// generators/relations

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