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G = D5×C40order 400 = 24·52

Direct product of C40 and D5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: D5×C40, C403C10, D10.4C20, C20.64D10, Dic5.4C20, C53(C2×C40), (C5×C40)⋊5C2, C52C86C10, C5215(C2×C8), C2.1(D5×C20), C10.8(C2×C20), (C4×D5).7C10, C4.12(D5×C10), C10.28(C4×D5), C20.13(C2×C10), (D5×C20).14C2, (D5×C10).12C4, (C5×C20).42C22, (C5×Dic5).12C4, (C5×C52C8)⋊13C2, (C5×C10).50(C2×C4), SmallGroup(400,76)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C40
C1C5C10C20C5×C20D5×C20 — D5×C40
C5 — D5×C40
C1C40

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

5C2
5C2
2C5
2C5
5C22
5C4
2C10
2C10
5C10
5C10
5C8
5C2×C4
2C20
2C20
5C2×C10
5C20
5C2×C8
2C40
2C40
5C40
5C2×C20
5C2×C40

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

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

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

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

160 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D5E···5N8A8B8C8D8E8F8G8H10A10B10C10D10E···10N10O···10V20A···20H20I···20AB20AC···20AJ40A···40P40Q···40BD40BE···40BT
order1222444455555···5888888881010101010···1010···1020···2020···2020···2040···4040···4040···40
size1155115511112···21111555511112···25···51···12···25···51···12···25···5

160 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C4C4C5C8C10C10C10C20C20C40D5D10C4×D5C5×D5C8×D5D5×C10D5×C20D5×C40
kernelD5×C40C5×C52C8C5×C40D5×C20C5×Dic5D5×C10C8×D5C5×D5C52C8C40C4×D5Dic5D10D5C40C20C10C8C5C4C2C1
# reps1111224844488322248881632

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

130
013
,
370
3010
,
319
3010
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

Export

Subgroup lattice of D5×C40 in TeX

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