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G = C5×C8.26D4order 320 = 26·5

Direct product of C5 and C8.26D4

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

Aliases: C5×C8.26D4, D84C20, Q164C20, SD162C20, C40.105D4, C4≀C26C10, C8○D47C10, (C5×D8)⋊16C4, C8.6(C2×C20), C8⋊C43C10, C8.25(C5×D4), C40.88(C2×C4), (C5×Q16)⋊16C4, C4○D8.3C10, D4.4(C2×C20), C4.83(D4×C10), C2.19(D4×C20), Q8.4(C2×C20), C8.C44C10, (C5×SD16)⋊10C4, C10.151(C4×D4), C20.488(C2×D4), C42.11(C2×C10), C4.16(C22×C20), (C2×C20).911C23, (C4×C20).252C22, (C2×C40).271C22, C20.220(C22×C4), M4(2).12(C2×C10), (C5×M4(2)).46C22, (C5×C4≀C2)⋊14C2, (C5×C8○D4)⋊16C2, (C5×C4○D8).8C2, (C5×C8⋊C4)⋊12C2, (C2×C8).54(C2×C10), C4○D4.9(C2×C10), (C5×D4).35(C2×C4), (C5×Q8).37(C2×C4), (C5×C8.C4)⋊13C2, C22.2(C5×C4○D4), (C2×C10).50(C4○D4), (C2×C4).86(C22×C10), (C5×C4○D4).54C22, SmallGroup(320,945)

Series: Derived Chief Lower central Upper central

C1C4 — C5×C8.26D4
C1C2C4C2×C4C2×C20C5×M4(2)C5×C4≀C2 — C5×C8.26D4
C1C2C4 — C5×C8.26D4
C1C20C2×C40 — C5×C8.26D4

Generators and relations for C5×C8.26D4
 G = < a,b,c,d | a5=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >

Subgroups: 154 in 104 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×3], C22, C22 [×2], C5, C8 [×2], C8 [×2], 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 [×3], C2×C10, C2×C10 [×2], C8⋊C4, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C40 [×2], C40 [×2], C40 [×2], C2×C20, C2×C20 [×3], C5×D4 [×2], C5×D4 [×2], C5×Q8 [×2], C8.26D4, 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], C5×C8⋊C4, C5×C4≀C2 [×2], C5×C8.C4, C5×C8○D4 [×2], C5×C4○D8, C5×C8.26D4
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, C8.26D4, C22×C20, D4×C10, C5×C4○D4, D4×C20, C5×C8.26D4

Smallest permutation representation of C5×C8.26D4
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)
(2 6)(4 8)(10 14)(12 16)(17 19 21 23)(18 24 22 20)(26 30)(28 32)(34 38)(36 40)(42 46)(44 48)(49 55 53 51)(50 52 54 56)(57 59 61 63)(58 64 62 60)(65 71 69 67)(66 68 70 72)(73 75 77 79)(74 80 78 76)
(1 59 3 61 5 63 7 57)(2 64 4 58 6 60 8 62)(9 56 11 50 13 52 15 54)(10 53 12 55 14 49 16 51)(17 47 19 41 21 43 23 45)(18 44 20 46 22 48 24 42)(25 70 27 72 29 66 31 68)(26 67 28 69 30 71 32 65)(33 73 35 75 37 77 39 79)(34 78 36 80 38 74 40 76)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G5A5B5C5D8A8B8C8D8E···8J10A10B10C10D10E10F10G10H10I···10P20A···20H20I20J20K20L20M···20AB40A···40P40Q···40AN
order122224444444555588888···8101010101010101010···1020···202020202020···2040···4040···40
size112441124444111122224···4111122224···41···122224···42···24···4

110 irreducible representations

dim111111111111111111222244
type+++++++
imageC1C2C2C2C2C2C4C4C4C5C10C10C10C10C10C20C20C20D4C4○D4C5×D4C5×C4○D4C8.26D4C5×C8.26D4
kernelC5×C8.26D4C5×C8⋊C4C5×C4≀C2C5×C8.C4C5×C8○D4C5×C4○D8C5×D8C5×SD16C5×Q16C8.26D4C8⋊C4C4≀C2C8.C4C8○D4C4○D8D8SD16Q16C40C2×C10C8C22C5C1
# reps1121212424484848168228828

Matrix representation of C5×C8.26D4 in GL4(𝔽41) generated by

37000
03700
00370
00037
,
16161239
23252532
0001
0090
,
12025
04000
0090
00032
,
20161118
00032
39212140
04000
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[16,23,0,0,16,25,0,0,12,25,0,9,39,32,1,0],[1,0,0,0,20,40,0,0,2,0,9,0,5,0,0,32],[20,0,39,0,16,0,21,40,11,0,21,0,18,32,40,0] >;

C5×C8.26D4 in GAP, Magma, Sage, TeX

C_5\times C_8._{26}D_4
% in TeX

G:=Group("C5xC8.26D4");
// GroupNames label

G:=SmallGroup(320,945);
// by ID

G=gap.SmallGroup(320,945);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,436,7004,3511,172,124]);
// Polycyclic

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

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