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## G = C6×D16order 192 = 26·3

### Direct product of C6 and D16

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C6×D16
 Chief series C1 — C2 — C4 — C8 — C24 — C3×D8 — C3×D16 — C6×D16
 Lower central C1 — C2 — C4 — C8 — C6×D16
 Upper central C1 — C2×C6 — C2×C12 — C2×C24 — C6×D16

Generators and relations for C6×D16
G = < a,b,c | a6=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 274 in 98 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, D4, C23, C12, C2×C6, C2×C6, C16, C2×C8, D8, D8, C2×D4, C24, C2×C12, C3×D4, C22×C6, C2×C16, D16, C2×D8, C48, C2×C24, C3×D8, C3×D8, C6×D4, C2×D16, C2×C48, C3×D16, C6×D8, C6×D16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C3×D4, C22×C6, D16, C2×D8, C3×D8, C6×D4, C2×D16, C3×D16, C6×D8, C6×D16

Smallest permutation representation of C6×D16
On 96 points
Generators in S96
(1 79 35 61 21 89)(2 80 36 62 22 90)(3 65 37 63 23 91)(4 66 38 64 24 92)(5 67 39 49 25 93)(6 68 40 50 26 94)(7 69 41 51 27 95)(8 70 42 52 28 96)(9 71 43 53 29 81)(10 72 44 54 30 82)(11 73 45 55 31 83)(12 74 46 56 32 84)(13 75 47 57 17 85)(14 76 48 58 18 86)(15 77 33 59 19 87)(16 78 34 60 20 88)
(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)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 52)(2 51)(3 50)(4 49)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 82)(34 81)(35 96)(36 95)(37 94)(38 93)(39 92)(40 91)(41 90)(42 89)(43 88)(44 87)(45 86)(46 85)(47 84)(48 83)

G:=sub<Sym(96)| (1,79,35,61,21,89)(2,80,36,62,22,90)(3,65,37,63,23,91)(4,66,38,64,24,92)(5,67,39,49,25,93)(6,68,40,50,26,94)(7,69,41,51,27,95)(8,70,42,52,28,96)(9,71,43,53,29,81)(10,72,44,54,30,82)(11,73,45,55,31,83)(12,74,46,56,32,84)(13,75,47,57,17,85)(14,76,48,58,18,86)(15,77,33,59,19,87)(16,78,34,60,20,88), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,52)(2,51)(3,50)(4,49)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,82)(34,81)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83)>;

G:=Group( (1,79,35,61,21,89)(2,80,36,62,22,90)(3,65,37,63,23,91)(4,66,38,64,24,92)(5,67,39,49,25,93)(6,68,40,50,26,94)(7,69,41,51,27,95)(8,70,42,52,28,96)(9,71,43,53,29,81)(10,72,44,54,30,82)(11,73,45,55,31,83)(12,74,46,56,32,84)(13,75,47,57,17,85)(14,76,48,58,18,86)(15,77,33,59,19,87)(16,78,34,60,20,88), (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)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,52)(2,51)(3,50)(4,49)(5,64)(6,63)(7,62)(8,61)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,82)(34,81)(35,96)(36,95)(37,94)(38,93)(39,92)(40,91)(41,90)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84)(48,83) );

G=PermutationGroup([[(1,79,35,61,21,89),(2,80,36,62,22,90),(3,65,37,63,23,91),(4,66,38,64,24,92),(5,67,39,49,25,93),(6,68,40,50,26,94),(7,69,41,51,27,95),(8,70,42,52,28,96),(9,71,43,53,29,81),(10,72,44,54,30,82),(11,73,45,55,31,83),(12,74,46,56,32,84),(13,75,47,57,17,85),(14,76,48,58,18,86),(15,77,33,59,19,87),(16,78,34,60,20,88)], [(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),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,52),(2,51),(3,50),(4,49),(5,64),(6,63),(7,62),(8,61),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,82),(34,81),(35,96),(36,95),(37,94),(38,93),(39,92),(40,91),(41,90),(42,89),(43,88),(44,87),(45,86),(46,85),(47,84),(48,83)]])

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 8A 8B 8C 8D 12A 12B 12C 12D 16A ··· 16H 24A ··· 24H 48A ··· 48P order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 16 ··· 16 24 ··· 24 48 ··· 48 size 1 1 1 1 8 8 8 8 1 1 2 2 1 ··· 1 8 ··· 8 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 D8 D8 C3×D4 C3×D4 D16 C3×D8 C3×D8 C3×D16 kernel C6×D16 C2×C48 C3×D16 C6×D8 C2×D16 C2×C16 D16 C2×D8 C24 C2×C12 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 1 4 2 2 2 8 4 1 1 2 2 2 2 8 4 4 16

Matrix representation of C6×D16 in GL4(𝔽97) generated by

 96 0 0 0 0 35 0 0 0 0 1 0 0 0 0 1
,
 96 0 0 0 0 96 0 0 0 0 26 95 0 0 2 26
,
 96 0 0 0 0 1 0 0 0 0 71 2 0 0 2 26
G:=sub<GL(4,GF(97))| [96,0,0,0,0,35,0,0,0,0,1,0,0,0,0,1],[96,0,0,0,0,96,0,0,0,0,26,2,0,0,95,26],[96,0,0,0,0,1,0,0,0,0,71,2,0,0,2,26] >;

C6×D16 in GAP, Magma, Sage, TeX

C_6\times D_{16}
% in TeX

G:=Group("C6xD16");
// GroupNames label

G:=SmallGroup(192,938);
// by ID

G=gap.SmallGroup(192,938);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,2524,1271,242,6053,3036,124]);
// Polycyclic

G:=Group<a,b,c|a^6=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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