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

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

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

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

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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