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G = D40⋊C4order 320 = 26·5

2nd semidirect product of D40 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D83F5, D402C4, C82(C2×F5), C401(C2×C4), D4⋊D52C4, C5⋊(D8⋊C4), (C5×D8)⋊2C4, D42(C2×F5), (D4×F5)⋊2C2, D202(C2×C4), C40⋊C43C2, C8⋊F51C2, (D5×D8).3C2, (C2×F5).5D4, C2.16(D4×F5), D20⋊C42C2, C10.15(C4×D4), C4⋊F5.2C22, C4.2(C22×F5), D10.64(C2×D4), C20.2(C22×C4), D5⋊C8.1C22, (D4×D5).6C22, (C4×F5).1C22, D5.2(C8⋊C22), (C8×D5).13C22, (C4×D5).24C23, Dic5.2(C4○D4), (C5×D4)⋊2(C2×C4), C52C812(C2×C4), SmallGroup(320,1069)

Series: Derived Chief Lower central Upper central

C1C20 — D40⋊C4
C1C5C10D10C4×D5C4×F5D4×F5 — D40⋊C4
C5C10C20 — D40⋊C4
C1C2C4D8

Generators and relations for D40⋊C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a13, cbc-1=a32b >

Subgroups: 674 in 132 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×6], C4, C4 [×5], C22 [×9], C5, C8, C8 [×2], C2×C4 [×9], D4 [×2], D4 [×4], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8, D8 [×3], C22×C4 [×2], C2×D4 [×2], Dic5, C20, F5 [×4], D10, D10 [×6], C2×C10 [×2], C8⋊C4, D4⋊C4 [×2], C4.Q8, C4×D4 [×2], C2×D8, C52C8, C40, C5⋊C8, C4×D5, D20 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5 [×2], C2×F5 [×6], C22×D5 [×2], D8⋊C4, C8×D5, D40, D4⋊D5 [×2], C5×D8, D5⋊C8, C4×F5, C4⋊F5 [×2], C22⋊F5 [×2], D4×D5 [×2], C22×F5 [×2], C8⋊F5, C40⋊C4, D20⋊C4 [×2], D5×D8, D4×F5 [×2], D40⋊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], D8⋊C4, C22×F5, D4×F5, D40⋊C4

Character table of D40⋊C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J58A8B8C8D10A10B10C2040A40B
 size 114455202021010101010202020204420202041616888
ρ111111111111111111111111111111    trivial
ρ211-1-111-1-1111111-1-1-1-1111111-1-1111    linear of order 2
ρ3111-1111-111-1-1-1-1-111-11-111-111-11-1-1    linear of order 2
ρ411-1111-1111-1-1-1-11-1-111-111-11-111-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-111-1-11111111    linear of order 2
ρ611-1-111-1-111-1-1-1-1111111-1-111-1-1111    linear of order 2
ρ7111-1111-11111111-1-111-1-1-1-111-11-1-1    linear of order 2
ρ811-1111-11111111-111-11-1-1-1-11-111-1-1    linear of order 2
ρ911-11-1-11-11-1ii-i-iii-i-i1-1i-i11-111-1-1    linear of order 4
ρ10111-1-1-1-111-1ii-i-i-i-iii1-1i-i111-11-1-1    linear of order 4
ρ11111-1-1-1-111-1-i-iiiii-i-i1-1-ii111-11-1-1    linear of order 4
ρ1211-11-1-11-11-1-i-iii-i-iii1-1-ii11-111-1-1    linear of order 4
ρ131111-1-1-1-11-1ii-i-i-ii-ii11-ii-1111111    linear of order 4
ρ1411-1-1-1-1111-1ii-i-ii-ii-i11-ii-11-1-1111    linear of order 4
ρ1511-1-1-1-1111-1-i-iii-ii-ii11i-i-11-1-1111    linear of order 4
ρ161111-1-1-1-11-1-i-iiii-ii-i11i-i-1111111    linear of order 4
ρ1722002200-2-2-22-22000020000200-200    orthogonal lifted from D4
ρ1822002200-2-22-22-2000020000200-200    orthogonal lifted from D4
ρ192200-2-200-22-2i2i2i-2i000020000200-200    complex lifted from C4○D4
ρ202200-2-200-222i-2i-2i2i000020000200-200    complex lifted from C4○D4
ρ2144-4-400004000000000-14000-111-1-1-1    orthogonal lifted from C2×F5
ρ224-400-4400000000000040000-400000    orthogonal lifted from C8⋊C22
ρ23444-400004000000000-1-4000-1-11-111    orthogonal lifted from C2×F5
ρ24444400004000000000-14000-1-1-1-1-1-1    orthogonal lifted from F5
ρ254-4004-400000000000040000-400000    orthogonal lifted from C8⋊C22
ρ2644-4400004000000000-1-4000-11-1-111    orthogonal lifted from C2×F5
ρ2788000000-8000000000-20000-200200    orthogonal lifted from D4×F5
ρ288-80000000000000000-200002000-1010    orthogonal faithful
ρ298-80000000000000000-20000200010-10    orthogonal faithful

Smallest permutation representation of D40⋊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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)
(1 21)(2 18 10 34)(3 15 19 7)(4 12 28 20)(5 9 37 33)(8 40 24 32)(11 31)(13 25 29 17)(14 22 38 30)(23 35 39 27)

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

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

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

Matrix representation of D40⋊C4 in GL8(𝔽41)

250500000
41020000
3111600000
384031400000
00000001
000040404040
00001000
00000100
,
15000000
040000000
016100000
323140400000
000000040
000000400
000004000
000040000
,
400000000
040000000
100100000
41010000
00001000
00000001
00000100
000040404040

G:=sub<GL(8,GF(41))| [25,4,31,38,0,0,0,0,0,1,1,40,0,0,0,0,5,0,16,31,0,0,0,0,0,2,0,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,0,0,0,0,1,40,0,0],[1,0,0,32,0,0,0,0,5,40,16,31,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0],[40,0,10,4,0,0,0,0,0,40,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40] >;

D40⋊C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes C_4
% in TeX

G:=Group("D40:C4");
// GroupNames label

G:=SmallGroup(320,1069);
// by ID

G=gap.SmallGroup(320,1069);
# by ID

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

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^32*b>;
// generators/relations

Export

Character table of D40⋊C4 in TeX

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