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G = D5xC5:C8order 400 = 24·52

Direct product of D5 and C5:C8

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

Aliases: D5xC5:C8, D10.4F5, Dic5.5D10, C5:2(C8xD5), (C5xD5):1C8, C52:2(C2xC8), C2.2(D5xF5), C10.4(C4xD5), C52:3C8:1C2, (D5xC10).1C4, C10.31(C2xF5), C52:6C4.2C4, (D5xDic5).2C2, (C5xDic5).6C22, C5:4(C2xC5:C8), (C5xC5:C8):1C2, (C5xC10).4(C2xC4), SmallGroup(400,120)

Series: Derived Chief Lower central Upper central

C1C52 — D5xC5:C8
C1C5C52C5xC10C5xDic5C5xC5:C8 — D5xC5:C8
C52 — D5xC5:C8
C1C2

Generators and relations for D5xC5:C8
 G = < a,b,c,d | a5=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 240 in 47 conjugacy classes, 22 normal (18 characteristic)
Quotients: C1, C2, C4, C22, C8, C2xC4, D5, C2xC8, F5, D10, C5:C8, C4xD5, C2xF5, C8xD5, C2xC5:C8, D5xF5, D5xC5:C8
5C2
5C2
4C5
5C22
5C4
25C4
4C10
5C10
5C10
5C8
25C8
25C2xC4
5Dic5
5Dic5
5C20
5C2xC10
20Dic5
25C2xC8
5C2xDic5
5C4xD5
5C5:2C8
5C5:C8
5C40
5C8xD5
5C2xC5:C8

Smallest permutation representation of D5xC5:C8
On 80 points
Generators in S80
(1 33 75 19 57)(2 34 76 20 58)(3 35 77 21 59)(4 36 78 22 60)(5 37 79 23 61)(6 38 80 24 62)(7 39 73 17 63)(8 40 74 18 64)(9 71 53 45 29)(10 72 54 46 30)(11 65 55 47 31)(12 66 56 48 32)(13 67 49 41 25)(14 68 50 42 26)(15 69 51 43 27)(16 70 52 44 28)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)
(1 33 75 19 57)(2 20 34 58 76)(3 59 21 77 35)(4 78 60 36 22)(5 37 79 23 61)(6 24 38 62 80)(7 63 17 73 39)(8 74 64 40 18)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)

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

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

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

40 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D5E8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G20A20B20C20D40A···40H
order122244445555588888888101010101010102020202040···40
size1155552525224885555252525252248820201010101010···10

40 irreducible representations

dim1111111222244488
type+++++++-++-
imageC1C2C2C2C4C4C8D5D10C4xD5C8xD5F5C5:C8C2xF5D5xF5D5xC5:C8
kernelD5xC5:C8C5xC5:C8C52:3C8D5xDic5C52:6C4D5xC10C5xD5C5:C8Dic5C10C5D10D5C10C2C1
# reps1111228224812122

Matrix representation of D5xC5:C8 in GL6(F41)

0400000
1340000
001000
000100
000010
000001
,
4070000
010000
001000
000100
000010
000001
,
100000
010000
000100
000010
0040404040
001000
,
300000
030000
001000
0040404040
000100
000010

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;

D5xC5:C8 in GAP, Magma, Sage, TeX

D_5\times C_5\rtimes C_8
% in TeX

G:=Group("D5xC5:C8");
// GroupNames label

G:=SmallGroup(400,120);
// by ID

G=gap.SmallGroup(400,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,970,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

Export

Subgroup lattice of D5xC5:C8 in TeX

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