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G = D85F5order 320 = 26·5

The semidirect product of D8 and F5 acting through Inn(D8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D85F5, Dic205C4, C5⋊C8.4D4, (C5×D8)⋊6C4, C51(C8○D8), (C8×F5)⋊2C2, D4.D51C4, D4.F51C2, D4.1(C2×F5), C2.17(D4×F5), C8.14(C2×F5), C40.12(C2×C4), D4⋊F51C2, C10.16(C4×D4), D10.Q83C2, C4.3(C22×F5), D83D5.5C2, C20.3(C22×C4), D10.1(C4○D4), C4.F5.1C22, D5⋊C8.10C22, Dic5.72(C2×D4), Dic10.1(C2×C4), (C4×D5).25C23, (C8×D5).22C22, (C4×F5).10C22, D42D5.4C22, C52C8.7(C2×C4), (C5×D4).1(C2×C4), SmallGroup(320,1070)

Series: Derived Chief Lower central Upper central

C1C20 — D85F5
C1C5C10Dic5C4×D5D5⋊C8D4.F5 — D85F5
C5C10C20 — D85F5
C1C2C4D8

Generators and relations for D85F5
 G = < a,b,c,d | a8=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 386 in 106 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2 [×3], C4, C4 [×5], 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 [×2], D10, C2×C10 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C52C8, C40, C5⋊C8 [×2], C5⋊C8 [×2], Dic10 [×2], C4×D5, C2×Dic5 [×2], C5⋊D4 [×2], C5×D4 [×2], C2×F5, C8○D8, C8×D5, Dic20, D4.D5 [×2], C5×D8, D5⋊C8, C4.F5 [×2], C4×F5, C2×C5⋊C8 [×2], C22.F5 [×2], D42D5 [×2], C8×F5, D10.Q8, D4⋊F5 [×2], D83D5, D4.F5 [×2], D85F5
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○D8, C22×F5, D4×F5, D85F5

Smallest permutation representation of D85F5
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 74)(2 73)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(49 70)(50 69)(51 68)(52 67)(53 66)(54 65)(55 72)(56 71)
(1 60 28 12 66)(2 61 29 13 67)(3 62 30 14 68)(4 63 31 15 69)(5 64 32 16 70)(6 57 25 9 71)(7 58 26 10 72)(8 59 27 11 65)(17 46 54 75 38)(18 47 55 76 39)(19 48 56 77 40)(20 41 49 78 33)(21 42 50 79 34)(22 43 51 80 35)(23 44 52 73 36)(24 45 53 74 37)
(9 57 25 71)(10 58 26 72)(11 59 27 65)(12 60 28 66)(13 61 29 67)(14 62 30 68)(15 63 31 69)(16 64 32 70)(17 56 42 36)(18 49 43 37)(19 50 44 38)(20 51 45 39)(21 52 46 40)(22 53 47 33)(23 54 48 34)(24 55 41 35)(73 75 77 79)(74 76 78 80)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I 5 8A8B8C8D8E8F8G8H8I8J8K8L8M8N10A10B10C 20 40A40B
order12222444444444588888888888888101010204040
size1144102551010101020204225555101010102020202041616888

35 irreducible representations

dim11111111122244488
type+++++++++++-
imageC1C2C2C2C2C2C4C4C4D4C4○D4C8○D8F5C2×F5C2×F5D4×F5D85F5
kernelD85F5C8×F5D10.Q8D4⋊F5D83D5D4.F5Dic20D4.D5C5×D8C5⋊C8D10C5D8C8D4C2C1
# reps11121224222811212

Matrix representation of D85F5 in GL6(𝔽41)

300000
34140000
001000
000100
000010
000001
,
14190000
7270000
0040000
0004000
0000400
0000040
,
100000
010000
0040404040
001000
000100
000010
,
100000
25320000
001000
000001
000100
0040404040

G:=sub<GL(6,GF(41))| [3,34,0,0,0,0,0,14,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,7,0,0,0,0,19,27,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[1,25,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40] >;

D85F5 in GAP, Magma, Sage, TeX

D_8\rtimes_5F_5
% in TeX

G:=Group("D8:5F5");
// GroupNames label

G:=SmallGroup(320,1070);
// by ID

G=gap.SmallGroup(320,1070);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,851,438,102,6278,1595]);
// 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,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

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