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G = C5×D82C4order 320 = 26·5

Direct product of C5 and D82C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×D82C4, D82C20, Q162C20, C40.102D4, M5(2)⋊5C10, C20.43SD16, (C5×D8)⋊14C4, C8.1(C2×C20), C4.Q81C10, C8.22(C5×D4), C40.83(C2×C4), (C5×Q16)⋊14C4, C4○D8.2C10, (C2×C10).25D8, C4.8(C5×SD16), C22.3(C5×D8), (C2×C20).279D4, (C5×M5(2))⋊13C2, (C2×C40).265C22, C10.55(D4⋊C4), C20.119(C22⋊C4), (C5×C4○D8).7C2, (C5×C4.Q8)⋊10C2, (C2×C4).10(C5×D4), C4.4(C5×C22⋊C4), (C2×C8).12(C2×C10), C2.9(C5×D4⋊C4), SmallGroup(320,165)

Series: Derived Chief Lower central Upper central

C1C8 — C5×D82C4
C1C2C4C2×C4C2×C8C2×C40C5×C4.Q8 — C5×D82C4
C1C2C4C8 — C5×D82C4
C1C10C2×C20C2×C40 — C5×D82C4

Generators and relations for C5×D82C4
 G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 130 in 58 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C10, C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, C20, C20, C2×C10, C2×C10, C4.Q8, M5(2), C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, D82C4, C80, C5×C4⋊C4, C2×C40, C5×D8, C5×SD16, C5×Q16, C5×C4○D4, C5×C4.Q8, C5×M5(2), C5×C4○D8, C5×D82C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, C2×C20, C5×D4, D82C4, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×D82C4

Smallest permutation representation of C5×D82C4
On 80 points
Generators in S80
(1 39 31 23 15)(2 40 32 24 16)(3 33 25 17 9)(4 34 26 18 10)(5 35 27 19 11)(6 36 28 20 12)(7 37 29 21 13)(8 38 30 22 14)(41 73 65 57 49)(42 74 66 58 50)(43 75 67 59 51)(44 76 68 60 52)(45 77 69 61 53)(46 78 70 62 54)(47 79 71 63 55)(48 80 72 64 56)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 56)(16 55)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 64)(24 63)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 72)(32 71)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 80)(40 79)
(2 4)(3 7)(6 8)(9 13)(10 16)(12 14)(17 21)(18 24)(20 22)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 42 45 46)(43 48 47 44)(49 50 53 54)(51 56 55 52)(57 58 61 62)(59 64 63 60)(65 66 69 70)(67 72 71 68)(73 74 77 78)(75 80 79 76)

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

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

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

80 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B5C5D8A8B8C10A10B10C10D10E10F10G10H10I10J10K10L16A16B16C16D20A···20H20I···20T40A···40H40I40J40K40L80A···80P
order12224444455558881010101010101010101010101616161620···2020···2040···404040404080···80
size112822888111122411112222888844442···28···82···244444···4

80 irreducible representations

dim1111111111112222222244
type+++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D4D4SD16D8C5×D4C5×D4C5×SD16C5×D8D82C4C5×D82C4
kernelC5×D82C4C5×C4.Q8C5×M5(2)C5×C4○D8C5×D8C5×Q16D82C4C4.Q8M5(2)C4○D8D8Q16C40C2×C20C20C2×C10C8C2×C4C4C22C5C1
# reps1111224444881122448828

Matrix representation of C5×D82C4 in GL4(𝔽241) generated by

87000
08700
00870
00087
,
0194332
2033816611
00222222
0019222
,
6411158121
0019222
203016611
2033816611
,
24001175
239119964
00222222
0022219
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[0,203,0,0,19,38,0,0,43,166,222,19,32,11,222,222],[64,0,203,203,11,0,0,38,158,19,166,166,121,222,11,11],[240,239,0,0,0,1,0,0,11,199,222,222,75,64,222,19] >;

C5×D82C4 in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes_2C_4
% in TeX

G:=Group("C5xD8:2C4");
// GroupNames label

G:=SmallGroup(320,165);
// by ID

G=gap.SmallGroup(320,165);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,3650,136,3511,10085,5052,124]);
// Polycyclic

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

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