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## G = D5×C40order 400 = 24·52

### Direct product of C40 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C40
 Chief series C1 — C5 — C10 — C20 — C5×C20 — D5×C20 — D5×C40
 Lower central C5 — D5×C40
 Upper central C1 — C40

Generators and relations for D5×C40
G = < a,b,c | a40=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C40
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 33 25 17 9)(2 34 26 18 10)(3 35 27 19 11)(4 36 28 20 12)(5 37 29 21 13)(6 38 30 22 14)(7 39 31 23 15)(8 40 32 24 16)(41 49 57 65 73)(42 50 58 66 74)(43 51 59 67 75)(44 52 60 68 76)(45 53 61 69 77)(46 54 62 70 78)(47 55 63 71 79)(48 56 64 72 80)
(1 79)(2 80)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 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), (1,33,25,17,9)(2,34,26,18,10)(3,35,27,19,11)(4,36,28,20,12)(5,37,29,21,13)(6,38,30,22,14)(7,39,31,23,15)(8,40,32,24,16)(41,49,57,65,73)(42,50,58,66,74)(43,51,59,67,75)(44,52,60,68,76)(45,53,61,69,77)(46,54,62,70,78)(47,55,63,71,79)(48,56,64,72,80), (1,79)(2,80)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)>;

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

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

160 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5N 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 20A ··· 20H 20I ··· 20AB 20AC ··· 20AJ 40A ··· 40P 40Q ··· 40BD 40BE ··· 40BT order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 8 8 8 8 8 8 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 40 ··· 40 size 1 1 5 5 1 1 5 5 1 1 1 1 2 ··· 2 1 1 1 1 5 5 5 5 1 1 1 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C5 C8 C10 C10 C10 C20 C20 C40 D5 D10 C4×D5 C5×D5 C8×D5 D5×C10 D5×C20 D5×C40 kernel D5×C40 C5×C5⋊2C8 C5×C40 D5×C20 C5×Dic5 D5×C10 C8×D5 C5×D5 C5⋊2C8 C40 C4×D5 Dic5 D10 D5 C40 C20 C10 C8 C5 C4 C2 C1 # reps 1 1 1 1 2 2 4 8 4 4 4 8 8 32 2 2 4 8 8 8 16 32

Matrix representation of D5×C40 in GL2(𝔽41) generated by

 13 0 0 13
,
 37 0 30 10
,
 31 9 30 10
G:=sub<GL(2,GF(41))| [13,0,0,13],[37,30,0,10],[31,30,9,10] >;

D5×C40 in GAP, Magma, Sage, TeX

D_5\times C_{40}
% in TeX

G:=Group("D5xC40");
// GroupNames label

G:=SmallGroup(400,76);
// by ID

G=gap.SmallGroup(400,76);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,127,69,11525]);
// Polycyclic

G:=Group<a,b,c|a^40=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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