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## G = C10×D20order 400 = 24·52

### Direct product of C10 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10×D20
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C2×C10 — C10×D20
 Lower central C5 — C10 — C10×D20
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C10×D20
G = < a,b,c | a10=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 388 in 124 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, C23, D5, C10, C10, C10, C2×D4, C20, C20, D10, D10, C2×C10, C2×C10, C52, D20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C5×D5, C5×C10, C5×C10, C2×D20, D4×C10, C5×C20, D5×C10, D5×C10, C102, C5×D20, C10×C20, D5×C2×C10, C10×D20
Quotients: C1, C2, C22, C5, D4, C23, D5, C10, C2×D4, D10, C2×C10, D20, C5×D4, C22×D5, C22×C10, C5×D5, C2×D20, D4×C10, D5×C10, C5×D20, D5×C2×C10, C10×D20

Smallest permutation representation of C10×D20
On 80 points
Generators in S80
(1 64 9 72 17 80 5 68 13 76)(2 65 10 73 18 61 6 69 14 77)(3 66 11 74 19 62 7 70 15 78)(4 67 12 75 20 63 8 71 16 79)(21 58 33 50 25 42 37 54 29 46)(22 59 34 51 26 43 38 55 30 47)(23 60 35 52 27 44 39 56 31 48)(24 41 36 53 28 45 40 57 32 49)
(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 44)(2 43)(3 42)(4 41)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 62)(22 61)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)

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

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

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

130 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 10AQ ··· 10BF 20A ··· 20AV order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 10 10 2 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10 2 ··· 2

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D5 D10 D10 D20 C5×D4 C5×D5 D5×C10 D5×C10 C5×D20 kernel C10×D20 C5×D20 C10×C20 D5×C2×C10 C2×D20 D20 C2×C20 C22×D5 C5×C10 C2×C20 C20 C2×C10 C10 C10 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 16 4 8 2 2 4 2 8 8 8 16 8 32

Matrix representation of C10×D20 in GL4(𝔽41) generated by

 23 0 0 0 0 23 0 0 0 0 16 0 0 0 0 16
,
 25 0 0 0 21 23 0 0 0 0 2 0 0 0 0 21
,
 6 17 0 0 10 35 0 0 0 0 0 21 0 0 2 0
G:=sub<GL(4,GF(41))| [23,0,0,0,0,23,0,0,0,0,16,0,0,0,0,16],[25,21,0,0,0,23,0,0,0,0,2,0,0,0,0,21],[6,10,0,0,17,35,0,0,0,0,0,2,0,0,21,0] >;

C10×D20 in GAP, Magma, Sage, TeX

C_{10}\times D_{20}
% in TeX

G:=Group("C10xD20");
// GroupNames label

G:=SmallGroup(400,183);
// by ID

G=gap.SmallGroup(400,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,794,194,11525]);
// Polycyclic

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

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