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## G = C5×C8.Q8order 320 = 26·5

### Direct product of C5 and C8.Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×C8.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C4.Q8 — C5×C8.Q8
 Lower central C1 — C2 — C4 — C8 — C5×C8.Q8
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×C8.Q8

Generators and relations for C5×C8.Q8
G = < a,b,c,d | a5=b8=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b3, dcd-1=b4c3 >

Smallest permutation representation of C5×C8.Q8
On 80 points
Generators in S80
(1 55 70 20 44)(2 56 71 21 45)(3 57 72 22 46)(4 58 73 23 47)(5 59 74 24 48)(6 60 75 25 33)(7 61 76 26 34)(8 62 77 27 35)(9 63 78 28 36)(10 64 79 29 37)(11 49 80 30 38)(12 50 65 31 39)(13 51 66 32 40)(14 52 67 17 41)(15 53 68 18 42)(16 54 69 19 43)
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 27 21 31 25 19 29 23)(18 20 22 24 26 28 30 32)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 60 54 64 58 52 62 56)(65 75 69 79 73 67 77 71)(66 68 70 72 74 76 78 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)
(2 4 10 12)(3 7)(5 13)(6 16 14 8)(11 15)(17 27 25 19)(18 30)(21 23 29 31)(22 26)(24 32)(33 43 41 35)(34 46)(37 39 45 47)(38 42)(40 48)(49 53)(50 56 58 64)(51 59)(52 62 60 54)(57 61)(65 71 73 79)(66 74)(67 77 75 69)(68 80)(72 76)

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

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

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

80 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 8C 8D 8E 10A 10B 10C 10D 10E 10F 10G 10H 16A 16B 16C 16D 20A ··· 20H 20I ··· 20P 40A ··· 40H 40I 40J 40K 40L 40M ··· 40T 80A ··· 80P order 1 2 2 4 4 4 4 5 5 5 5 8 8 8 8 8 10 10 10 10 10 10 10 10 16 16 16 16 20 ··· 20 20 ··· 20 40 ··· 40 40 40 40 40 40 ··· 40 80 ··· 80 size 1 1 2 2 2 8 8 1 1 1 1 2 2 4 8 8 1 1 1 1 2 2 2 2 4 4 4 4 2 ··· 2 8 ··· 8 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + - + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 Q8 D4 SD16 SD16 C5×Q8 C5×D4 C5×SD16 C5×SD16 C8.Q8 C5×C8.Q8 kernel C5×C8.Q8 C5×C4.Q8 C5×C8.C4 C5×M5(2) C80 C8.Q8 C4.Q8 C8.C4 M5(2) C16 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×C8.Q8 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 205 0 0 0 0 205
,
 222 19 121 9 222 222 139 111 0 0 38 19 0 0 203 0
,
 120 120 1 181 102 102 0 171 203 0 0 120 38 38 0 19
,
 1 0 120 111 0 240 102 231 0 0 0 19 0 0 38 0
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,205,0,0,0,0,205],[222,222,0,0,19,222,0,0,121,139,38,203,9,111,19,0],[120,102,203,38,120,102,0,38,1,0,0,0,181,171,120,19],[1,0,0,0,0,240,0,0,120,102,0,38,111,231,19,0] >;

C5×C8.Q8 in GAP, Magma, Sage, TeX

C_5\times C_8.Q_8
% in TeX

G:=Group("C5xC8.Q8");
// GroupNames label

G:=SmallGroup(320,170);
// by ID

G=gap.SmallGroup(320,170);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,148,2803,136,3511,10085,124]);
// Polycyclic

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

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