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## G = D5×C16order 160 = 25·5

### Direct product of C16 and D5

Aliases: D5×C16, C804C2, D10.4C8, C8.19D10, Dic5.4C8, C40.19C22, C53(C2×C16), C52C166C2, C2.1(C8×D5), C52C8.7C4, (C4×D5).9C4, C4.16(C4×D5), C20.42(C2×C4), C10.10(C2×C8), (C8×D5).11C2, SmallGroup(160,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C16
 Chief series C1 — C5 — C10 — C20 — C40 — C8×D5 — D5×C16
 Lower central C5 — D5×C16
 Upper central C1 — C16

Generators and relations for D5×C16
G = < a,b,c | a16=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C16
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 20 42 60 68)(2 21 43 61 69)(3 22 44 62 70)(4 23 45 63 71)(5 24 46 64 72)(6 25 47 49 73)(7 26 48 50 74)(8 27 33 51 75)(9 28 34 52 76)(10 29 35 53 77)(11 30 36 54 78)(12 31 37 55 79)(13 32 38 56 80)(14 17 39 57 65)(15 18 40 58 66)(16 19 41 59 67)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 72)(14 73)(15 74)(16 75)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)

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

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

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

D5×C16 is a maximal subgroup of
C32⋊D5  D5⋊C32  C80.C4  C167F5  C802C4  C803C4  C16.F5  C80.2C4  D20.6C8  D20.5C8  D163D5  SD323D5  D805C2  D152C16
D5×C16 is a maximal quotient of
C32⋊D5  C40.88D4  D101C16  D152C16

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 16A ··· 16H 16I ··· 16P 20A 20B 20C 20D 40A ··· 40H 80A ··· 80P order 1 2 2 2 4 4 4 4 5 5 8 8 8 8 8 8 8 8 10 10 16 ··· 16 16 ··· 16 20 20 20 20 40 ··· 40 80 ··· 80 size 1 1 5 5 1 1 5 5 2 2 1 1 1 1 5 5 5 5 2 2 1 ··· 1 5 ··· 5 2 2 2 2 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 D5 D10 C4×D5 C8×D5 D5×C16 kernel D5×C16 C5⋊2C16 C80 C8×D5 C5⋊2C8 C4×D5 Dic5 D10 D5 C16 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 2 2 4 8 16

Matrix representation of D5×C16 in GL2(𝔽241) generated by

 111 0 0 111
,
 189 1 240 0
,
 0 240 240 0
G:=sub<GL(2,GF(241))| [111,0,0,111],[189,240,1,0],[0,240,240,0] >;

D5×C16 in GAP, Magma, Sage, TeX

D_5\times C_{16}
% in TeX

G:=Group("D5xC16");
// GroupNames label

G:=SmallGroup(160,4);
// by ID

G=gap.SmallGroup(160,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,31,50,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^16=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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