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## G = D5×C2×C8order 160 = 25·5

### Direct product of C2×C8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C8
 Chief series C1 — C5 — C10 — C20 — C4×D5 — C2×C4×D5 — D5×C2×C8
 Lower central C5 — D5×C2×C8
 Upper central C1 — C2×C8

Generators and relations for D5×C2×C8
G = < a,b,c,d | a2=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 184 in 76 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C22×C8, C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C8×D5, C2×C52C8, C2×C40, C2×C4×D5, D5×C2×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C2×C8, C22×C4, D10, C22×C8, C4×D5, C22×D5, C8×D5, C2×C4×D5, D5×C2×C8

Smallest permutation representation of D5×C2×C8
On 80 points
Generators in S80
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)
(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 59 51 42 65)(2 60 52 43 66)(3 61 53 44 67)(4 62 54 45 68)(5 63 55 46 69)(6 64 56 47 70)(7 57 49 48 71)(8 58 50 41 72)(9 17 33 25 74)(10 18 34 26 75)(11 19 35 27 76)(12 20 36 28 77)(13 21 37 29 78)(14 22 38 30 79)(15 23 39 31 80)(16 24 40 32 73)
(1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)(25 29)(26 30)(27 31)(28 32)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)(49 53)(50 54)(51 55)(52 56)

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10F 20A ··· 20H 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 5 5 5 5 1 1 1 1 5 5 5 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D5 D10 D10 C4×D5 C4×D5 C8×D5 kernel D5×C2×C8 C8×D5 C2×C5⋊2C8 C2×C40 C2×C4×D5 C4×D5 C2×Dic5 C22×D5 D10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 16 2 4 2 4 4 16

Matrix representation of D5×C2×C8 in GL3(𝔽41) generated by

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

D5×C2×C8 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_8
% in TeX

G:=Group("D5xC2xC8");
// GroupNames label

G:=SmallGroup(160,120);
// by ID

G=gap.SmallGroup(160,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,50,69,4613]);
// Polycyclic

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

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