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## G = C10×C4.D4order 320 = 26·5

### Direct product of C10 and C4.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C10×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×M4(2) — C5×C4.D4 — C10×C4.D4
 Lower central C1 — C2 — C22 — C10×C4.D4
 Upper central C1 — C2×C10 — C22×C20 — C10×C4.D4

Generators and relations for C10×C4.D4
G = < a,b,c,d | a10=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 370 in 186 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C4.D4, C2×M4(2), C22×D4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C2×C4.D4, C2×C40, C5×M4(2), C5×M4(2), C22×C20, D4×C10, D4×C10, C23×C10, C5×C4.D4, C10×M4(2), D4×C2×C10, C10×C4.D4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C4.D4, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C2×C4.D4, C5×C22⋊C4, C22×C20, D4×C10, C5×C4.D4, C10×C22⋊C4, C10×C4.D4

Smallest permutation representation of C10×C4.D4
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 56 70 50)(2 57 61 41)(3 58 62 42)(4 59 63 43)(5 60 64 44)(6 51 65 45)(7 52 66 46)(8 53 67 47)(9 54 68 48)(10 55 69 49)(11 80 29 33)(12 71 30 34)(13 72 21 35)(14 73 22 36)(15 74 23 37)(16 75 24 38)(17 76 25 39)(18 77 26 40)(19 78 27 31)(20 79 28 32)
(1 74 56 15 70 37 50 23)(2 75 57 16 61 38 41 24)(3 76 58 17 62 39 42 25)(4 77 59 18 63 40 43 26)(5 78 60 19 64 31 44 27)(6 79 51 20 65 32 45 28)(7 80 52 11 66 33 46 29)(8 71 53 12 67 34 47 30)(9 72 54 13 68 35 48 21)(10 73 55 14 69 36 49 22)
(1 28 56 32 70 20 50 79)(2 29 57 33 61 11 41 80)(3 30 58 34 62 12 42 71)(4 21 59 35 63 13 43 72)(5 22 60 36 64 14 44 73)(6 23 51 37 65 15 45 74)(7 24 52 38 66 16 46 75)(8 25 53 39 67 17 47 76)(9 26 54 40 68 18 48 77)(10 27 55 31 69 19 49 78)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 5A 5B 5C 5D 8A ··· 8H 10A ··· 10L 10M ··· 10T 10U ··· 10AJ 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 1 1 1 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 C5×D4 C4.D4 C5×C4.D4 kernel C10×C4.D4 C5×C4.D4 C10×M4(2) D4×C2×C10 D4×C10 C23×C10 C2×C4.D4 C4.D4 C2×M4(2) C22×D4 C2×D4 C24 C2×C20 C2×C4 C10 C2 # reps 1 4 2 1 4 4 4 16 8 4 16 16 4 16 2 8

Matrix representation of C10×C4.D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,40,0,0] >;

C10×C4.D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4.D_4
% in TeX

G:=Group("C10xC4.D4");
// GroupNames label

G:=SmallGroup(320,912);
// by ID

G=gap.SmallGroup(320,912);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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