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## G = Dic5×C20order 400 = 24·52

### Direct product of C20 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C20
 Chief series C1 — C5 — C10 — C2×C10 — C102 — C10×Dic5 — Dic5×C20
 Lower central C5 — Dic5×C20
 Upper central C1 — C2×C20

Generators and relations for Dic5×C20
G = < a,b,c | a20=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 148 in 76 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, C2×C4, C10, C10, C10, C42, Dic5, C20, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C2×C20, C5×C10, C5×C10, C4×Dic5, C4×C20, C5×Dic5, C5×C20, C102, C10×Dic5, C10×C20, Dic5×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D5, C10, C42, Dic5, C20, D10, C2×C10, C4×D5, C2×Dic5, C2×C20, C5×D5, C4×Dic5, C4×C20, C5×Dic5, D5×C10, D5×C20, C10×Dic5, Dic5×C20

Smallest permutation representation of Dic5×C20
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 32 13 24 5 36 17 28 9 40)(2 33 14 25 6 37 18 29 10 21)(3 34 15 26 7 38 19 30 11 22)(4 35 16 27 8 39 20 31 12 23)(41 80 49 68 57 76 45 64 53 72)(42 61 50 69 58 77 46 65 54 73)(43 62 51 70 59 78 47 66 55 74)(44 63 52 71 60 79 48 67 56 75)
(1 53 36 68)(2 54 37 69)(3 55 38 70)(4 56 39 71)(5 57 40 72)(6 58 21 73)(7 59 22 74)(8 60 23 75)(9 41 24 76)(10 42 25 77)(11 43 26 78)(12 44 27 79)(13 45 28 80)(14 46 29 61)(15 47 30 62)(16 48 31 63)(17 49 32 64)(18 50 33 65)(19 51 34 66)(20 52 35 67)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 20A ··· 20P 20Q ··· 20BD 20BE ··· 20CJ order 1 2 2 2 4 4 4 4 4 ··· 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 1 1 1 1 1 1 5 ··· 5 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D5 Dic5 D10 C4×D5 C5×D5 C5×Dic5 D5×C10 D5×C20 kernel Dic5×C20 C10×Dic5 C10×C20 C5×Dic5 C5×C20 C4×Dic5 C2×Dic5 C2×C20 Dic5 C20 C2×C20 C20 C2×C10 C10 C2×C4 C4 C22 C2 # reps 1 2 1 8 4 4 8 4 32 16 2 4 2 8 8 16 8 32

Matrix representation of Dic5×C20 in GL3(𝔽41) generated by

 18 0 0 0 8 0 0 0 8
,
 40 0 0 0 10 0 0 0 37
,
 9 0 0 0 0 9 0 32 0
G:=sub<GL(3,GF(41))| [18,0,0,0,8,0,0,0,8],[40,0,0,0,10,0,0,0,37],[9,0,0,0,0,32,0,9,0] >;

Dic5×C20 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_{20}
% in TeX

G:=Group("Dic5xC20");
// GroupNames label

G:=SmallGroup(400,83);
// by ID

G=gap.SmallGroup(400,83);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,11525]);
// Polycyclic

G:=Group<a,b,c|a^20=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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