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G = C5×C8○D8order 320 = 26·5

Direct product of C5 and C8○D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C8○D8, D85C20, Q165C20, C40.77D4, SD163C20, C4≀C27C10, (C4×C40)⋊26C2, (C4×C8)⋊10C10, C8○D46C10, (C5×D8)⋊17C4, C8.30(C5×D4), C8.11(C2×C20), (C5×Q16)⋊17C4, C4○D8.5C10, D4.3(C2×C20), C4.82(D4×C10), C2.18(D4×C20), Q8.3(C2×C20), C8.C48C10, C40.108(C2×C4), (C5×SD16)⋊11C4, C20.487(C2×D4), C10.150(C4×D4), C42.73(C2×C10), C4.15(C22×C20), (C4×C20).358C22, (C2×C20).910C23, C20.219(C22×C4), (C2×C40).433C22, M4(2).11(C2×C10), (C5×M4(2)).45C22, (C5×C4≀C2)⋊15C2, (C5×C8○D4)⋊15C2, (C5×C4○D8).10C2, C4○D4.8(C2×C10), (C5×D4).34(C2×C4), (C5×Q8).36(C2×C4), (C5×C8.C4)⋊17C2, C22.1(C5×C4○D4), (C2×C8).101(C2×C10), (C2×C10).49(C4○D4), (C2×C4).85(C22×C10), (C5×C4○D4).53C22, SmallGroup(320,944)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C8○D8
C1C2C4C2×C4C2×C20C5×M4(2)C5×C4≀C2 — C5×C8○D8
C1C2C4 — C5×C8○D8
C1C40C2×C40 — C5×C8○D8

Generators and relations for C5×C8○D8
 G = < a,b,c,d | a5=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c3 >

Subgroups: 154 in 106 conjugacy classes, 66 normal (38 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×2], D4 [×2], Q8 [×2], C10, C10 [×3], C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×2], C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C40 [×4], C40 [×2], C2×C20, C2×C20 [×3], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C8○D8, C4×C20, C2×C40 [×2], C2×C40 [×2], C5×M4(2) [×2], C5×M4(2) [×2], C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C4×C40, C5×C4≀C2 [×2], C5×C8.C4, C5×C8○D4 [×2], C5×C4○D8, C5×C8○D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], C23, C10 [×7], C22×C4, C2×D4, C4○D4, C20 [×4], C2×C10 [×7], C4×D4, C2×C20 [×6], C5×D4 [×2], C22×C10, C8○D8, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8○D8

Smallest permutation representation of C5×C8○D8
On 80 points
Generators in S80
(1 43 25 39 11)(2 44 26 40 12)(3 45 27 33 13)(4 46 28 34 14)(5 47 29 35 15)(6 48 30 36 16)(7 41 31 37 9)(8 42 32 38 10)(17 72 73 52 61)(18 65 74 53 62)(19 66 75 54 63)(20 67 76 55 64)(21 68 77 56 57)(22 69 78 49 58)(23 70 79 50 59)(24 71 80 51 60)
(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 6 3 8 5 2 7 4)(9 14 11 16 13 10 15 12)(17 20 23 18 21 24 19 22)(25 30 27 32 29 26 31 28)(33 38 35 40 37 34 39 36)(41 46 43 48 45 42 47 44)(49 52 55 50 53 56 51 54)(57 60 63 58 61 64 59 62)(65 68 71 66 69 72 67 70)(73 76 79 74 77 80 75 78)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 45)(18 46)(19 47)(20 48)(21 41)(22 42)(23 43)(24 44)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

140 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I5A5B5C5D8A8B8C8D8E···8J8K8L8M8N10A10B10C10D10E10F10G10H10I···10P20A···20H20I···20AB20AC···20AJ40A···40P40Q···40AN40AO···40BD
order12222444···444555588888···88888101010101010101010···1020···2020···2020···2040···4040···4040···40
size11244112···244111111112···24444111122224···41···12···24···41···12···24···4

140 irreducible representations

dim111111111111111111222222
type+++++++
imageC1C2C2C2C2C2C4C4C4C5C10C10C10C10C10C20C20C20D4C4○D4C5×D4C8○D8C5×C4○D4C5×C8○D8
kernelC5×C8○D8C4×C40C5×C4≀C2C5×C8.C4C5×C8○D4C5×C4○D8C5×D8C5×SD16C5×Q16C8○D8C4×C8C4≀C2C8.C4C8○D4C4○D8D8SD16Q16C40C2×C10C8C5C22C1
# reps11212124244848481682288832

Matrix representation of C5×C8○D8 in GL2(𝔽41) generated by

180
018
,
140
014
,
270
038
,
01
10
G:=sub<GL(2,GF(41))| [18,0,0,18],[14,0,0,14],[27,0,0,38],[0,1,1,0] >;

C5×C8○D8 in GAP, Magma, Sage, TeX

C_5\times C_8\circ D_8
% in TeX

G:=Group("C5xC8oD8");
// GroupNames label

G:=SmallGroup(320,944);
// by ID

G=gap.SmallGroup(320,944);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,7004,3511,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^3>;
// generators/relations

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