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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, C4, C4, C22, C5, C8, C8, C2×C4, D4, D4, Q8, D5, C10, C10, C42, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C4×C8, C4≀C2, C8.C4, C8○D4, C4○D8, C52C8, C40, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C8○D8, C8×D5, Dic20, D4.D5, C5×D8, D5⋊C8, C4.F5, C4×F5, C2×C5⋊C8, C22.F5, D42D5, C8×F5, D10.Q8, D4⋊F5, D83D5, D4.F5, D85F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5, 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 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)
(9 28 65 76)(10 29 66 77)(11 30 67 78)(12 31 68 79)(13 32 69 80)(14 25 70 73)(15 26 71 74)(16 27 72 75)(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)(57 59 61 63)(58 60 62 64)

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), (9,28,65,76)(10,29,66,77)(11,30,67,78)(12,31,68,79)(13,32,69,80)(14,25,70,73)(15,26,71,74)(16,27,72,75)(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)(57,59,61,63)(58,60,62,64)>;

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), (9,28,65,76)(10,29,66,77)(11,30,67,78)(12,31,68,79)(13,32,69,80)(14,25,70,73)(15,26,71,74)(16,27,72,75)(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)(57,59,61,63)(58,60,62,64) );

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)], [(9,28,65,76),(10,29,66,77),(11,30,67,78),(12,31,68,79),(13,32,69,80),(14,25,70,73),(15,26,71,74),(16,27,72,75),(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),(57,59,61,63),(58,60,62,64)]])

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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