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## G = C6×C3⋊C16order 288 = 25·32

### Direct product of C6 and C3⋊C16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×C3⋊C16
 Chief series C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×C3⋊C16 — C6×C3⋊C16
 Lower central C3 — C6×C3⋊C16
 Upper central C1 — C2×C24

Generators and relations for C6×C3⋊C16
G = < a,b,c | a6=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 90 in 67 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C16 [×2], C2×C8, C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], C2×C12 [×2], C2×C12, C2×C16, C3×C12 [×2], C62, C3⋊C16 [×2], C48 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×C12, C2×C3⋊C16, C2×C48, C3×C3⋊C16 [×2], C6×C24, C6×C3⋊C16
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, Dic3 [×2], C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3, C3⋊C8 [×2], C24 [×2], C2×Dic3, C2×C12, C2×C16, C3×Dic3 [×2], S3×C6, C3⋊C16 [×2], C48 [×2], C2×C3⋊C8, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C2×C3⋊C16, C2×C48, C3×C3⋊C16 [×2], C6×C3⋊C8, C6×C3⋊C16

Smallest permutation representation of C6×C3⋊C16
On 96 points
Generators in S96
(1 25 45 87 71 64)(2 26 46 88 72 49)(3 27 47 89 73 50)(4 28 48 90 74 51)(5 29 33 91 75 52)(6 30 34 92 76 53)(7 31 35 93 77 54)(8 32 36 94 78 55)(9 17 37 95 79 56)(10 18 38 96 80 57)(11 19 39 81 65 58)(12 20 40 82 66 59)(13 21 41 83 67 60)(14 22 42 84 68 61)(15 23 43 85 69 62)(16 24 44 86 70 63)
(1 45 71)(2 72 46)(3 47 73)(4 74 48)(5 33 75)(6 76 34)(7 35 77)(8 78 36)(9 37 79)(10 80 38)(11 39 65)(12 66 40)(13 41 67)(14 68 42)(15 43 69)(16 70 44)(17 95 56)(18 57 96)(19 81 58)(20 59 82)(21 83 60)(22 61 84)(23 85 62)(24 63 86)(25 87 64)(26 49 88)(27 89 50)(28 51 90)(29 91 52)(30 53 92)(31 93 54)(32 55 94)
(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)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 8A ··· 8H 12A ··· 12H 12I ··· 12T 16A ··· 16P 24A ··· 24P 24Q ··· 24AN 48A ··· 48AF order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 1 1 1 1 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C8 C12 C12 C16 C24 C24 C48 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C3⋊C16 C3×C3⋊C8 C3×C3⋊C8 C3×C3⋊C16 kernel C6×C3⋊C16 C3×C3⋊C16 C6×C24 C2×C3⋊C16 C3×C24 C6×C12 C3⋊C16 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C12 C2×C6 C6 C2×C24 C24 C24 C2×C12 C2×C8 C12 C2×C6 C8 C8 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 4 4 4 4 16 8 8 32 1 1 1 1 2 2 2 2 2 2 8 4 4 16

Matrix representation of C6×C3⋊C16 in GL3(𝔽97) generated by

 36 0 0 0 36 0 0 0 36
,
 1 0 0 0 35 0 0 43 61
,
 96 0 0 0 47 60 0 9 50
G:=sub<GL(3,GF(97))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,35,43,0,0,61],[96,0,0,0,47,9,0,60,50] >;

C6×C3⋊C16 in GAP, Magma, Sage, TeX

C_6\times C_3\rtimes C_{16}
% in TeX

G:=Group("C6xC3:C16");
// GroupNames label

G:=SmallGroup(288,245);
// by ID

G=gap.SmallGroup(288,245);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^3=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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