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G = D8⋊F5  order 320 = 26·5

4th semidirect product of D8 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8⋊4F5, Dic20⋊2C4, C5⋊C8.1D4, (C5×D8)⋊3C4, C8.6(C2×F5), C40.5(C2×C4), D4.D5⋊2C4, C8⋊F5⋊2C2, D4.F5⋊2C2, D4.2(C2×F5), C2.18(D4×F5), C5⋊1(C8.26D4), D4⋊F5⋊2C2, C10.17(C4×D4), C40.C4⋊2C2, C4.4(C22×F5), D8⋊3D5.3C2, C20.4(C22×C4), D5⋊C8.2C22, (C4×F5).2C22, D10.2(C4○D4), C4.F5.2C22, Dic5.73(C2×D4), Dic10.2(C2×C4), (C4×D5).26C23, (C8×D5).14C22, D4⋊2D5.5C22, C5⋊2C8.8(C2×C4), (C5×D4).2(C2×C4), SmallGroup(320,1071)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — D8⋊F5
Chief series
C1 — C5 — C10 — Dic5 — C4×D5 — D5⋊C8 — D4.F5 — D8⋊F5
Lower central
C5 — C10 — C20 — D8⋊F5
Upper central
C1 — C2 — C4 — D8

Subgroups: 386 in 104 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2 [×3], C4, C4 [×4], C22 [×3], C5, C8, C8 [×5], C2×C4 [×4], D4 [×2], D4 [×2], Q8 [×2], D5, C10, C10 [×2], C42, C2×C8 [×4], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic5, Dic5 [×2], C20, F5, D10, C2×C10 [×2], C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C5⋊2C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5, C8.26D4, C8×D5, Dic20, D4.D5 [×2], C5×D8, D5⋊C8, C4.F5 [×2], C4×F5, C2×C5⋊C8 [×2], C22.F5 [×2], D4⋊2D5 [×2], C8⋊F5, C40.C4, D4⋊F5 [×2], D8⋊3D5, D4.F5 [×2], D8⋊F5

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5 [×3], C8.26D4, C22×F5, D4×F5, D8⋊F5

Generators and relations
 G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a2b, dcd-1=c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

10000000
01000000
00100000
00010000
00000001
00000010
000003200
00009000
,
400000000
040000000
004000000
000400000
00000010
00000001
00001000
00000100
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
10000000
00010000
01000000
404040400000
00001000
00000900
000000320
000000040

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40] >;
 

Character table of D8⋊F5

 class 12A2B2C2D4A4B4C4D4E4F4G58A8B8C8D8E8F8G8H8I8J10A10B10C2040A40B
 size 114410255202020204410101010202020202041616888
ρ111111111111111111111111111111    trivial
ρ2111-11111111-11-1-1-1-1-1-111-1-111-11-1-1    linear of order 2
ρ311-1-11111-1-1-1-111-1-1-1-1111111-1-1111    linear of order 2
ρ411-111111-1-1-111-11111-111-1-11-111-1-1    linear of order 2
ρ5111-11111-11-1-11-11111-1-1-11111-11-1-1    linear of order 2
ρ611111111-11-1111-1-1-1-11-1-1-1-1111111    linear of order 2
ρ711-1111111-1111-1-1-1-1-1-1-1-1111-111-1-1    linear of order 2
ρ811-1-111111-11-11111111-1-1-1-11-1-1111    linear of order 2
ρ91111-11-1-1i-1-i-111-iii-i-1-ii-ii111111    linear of order 4
ρ10111-1-11-1-1i-1-i11-1i-i-ii1-iii-i11-11-1-1    linear of order 4
ρ111111-11-1-1-i-1i-111i-i-ii-1i-ii-i111111    linear of order 4
ρ12111-1-11-1-1-i-1i11-1-iii-i1i-i-ii11-11-1-1    linear of order 4
ρ1311-11-11-1-1-i1i-11-1-iii-i1-iii-i1-111-1-1    linear of order 4
ρ1411-1-1-11-1-1-i1i111i-i-ii-1-ii-ii1-1-1111    linear of order 4
ρ1511-11-11-1-1i1-i-11-1i-i-ii1i-i-ii1-111-1-1    linear of order 4
ρ1611-1-1-11-1-1i1-i111-iii-i-1i-ii-i1-1-1111    linear of order 4
ρ172200-2-222000020-22-2200000200-200    orthogonal lifted from D4
ρ182200-2-2220000202-22-200000200-200    orthogonal lifted from D4
ρ1922002-2-2-20000202i2i2i2i00000200-200    complex lifted from C4○D4
ρ2022002-2-2-20000202i2i2i2i00000200-200    complex lifted from C4○D4
ρ2144-4-404000000-14000000000-111-1-1-1    orthogonal lifted from C2×F5
ρ2244-4404000000-1-4000000000-11-1-111    orthogonal lifted from C2×F5
ρ23444-404000000-1-4000000000-1-11-111    orthogonal lifted from C2×F5
ρ24444404000000-14000000000-1-1-1-1-1-1    orthogonal lifted from F5
ρ254-400004i4i000040000000000-400000    complex lifted from C8.26D4
ρ264-400004i4i000040000000000-400000    complex lifted from C8.26D4
ρ2788000-8000000-20000000000-200200    orthogonal lifted from D4×F5
ρ288-80000000000-200000000002000√10√10    symplectic faithful, Schur index 2
ρ298-80000000000-200000000002000√10√10    symplectic faithful, Schur index 2

In GAP, Magma, Sage, TeX 

D_8\rtimes F_5
 
% in TeX
 
G:=Group("D8:F5");
 
// GroupNames label
 
G:=SmallGroup(320,1071);
 
// by ID
 
G=gap.SmallGroup(320,1071);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,851,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
 
// generators/relations
 

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