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## G = C3×S3×Q16order 288 = 25·32

### Direct product of C3, S3 and Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×S3×Q16
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×S3×Q8 — C3×S3×Q16
 Lower central C3 — C6 — C12 — C3×S3×Q16
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×S3×Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 282 in 129 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], Q8 [×2], Q8 [×4], C32, Dic3, Dic3 [×2], C12 [×2], C12 [×10], D6, C2×C6, C2×C8, Q16, Q16 [×3], C2×Q8 [×2], C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6 [×2], Dic6 [×2], C4×S3, C4×S3 [×2], C2×C12 [×3], C3×Q8 [×4], C3×Q8 [×6], C2×Q16, C3×Dic3, C3×Dic3 [×2], C3×C12, C3×C12 [×2], S3×C6, S3×C8, Dic12, C3⋊Q16 [×2], C2×C24, C3×Q16 [×2], C3×Q16 [×4], S3×Q8 [×2], C6×Q8 [×2], C3×C3⋊C8, C3×C24, C3×Dic6 [×2], C3×Dic6 [×2], S3×C12, S3×C12 [×2], Q8×C32 [×2], S3×Q16, C6×Q16, S3×C24, C3×Dic12, C3×C3⋊Q16 [×2], C32×Q16, C3×S3×Q8 [×2], C3×S3×Q16
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], Q16 [×2], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C2×Q16, S3×C6 [×3], C3×Q16 [×2], S3×D4, C6×D4, S3×C2×C6, S3×Q16, C6×Q16, C3×S3×D4, C3×S3×Q16

Smallest permutation representation of C3×S3×Q16
On 96 points
Generators in S96
(1 29 54)(2 30 55)(3 31 56)(4 32 49)(5 25 50)(6 26 51)(7 27 52)(8 28 53)(9 94 64)(10 95 57)(11 96 58)(12 89 59)(13 90 60)(14 91 61)(15 92 62)(16 93 63)(17 75 70)(18 76 71)(19 77 72)(20 78 65)(21 79 66)(22 80 67)(23 73 68)(24 74 69)(33 81 42)(34 82 43)(35 83 44)(36 84 45)(37 85 46)(38 86 47)(39 87 48)(40 88 41)
(1 29 54)(2 30 55)(3 31 56)(4 32 49)(5 25 50)(6 26 51)(7 27 52)(8 28 53)(9 64 94)(10 57 95)(11 58 96)(12 59 89)(13 60 90)(14 61 91)(15 62 92)(16 63 93)(17 75 70)(18 76 71)(19 77 72)(20 78 65)(21 79 66)(22 80 67)(23 73 68)(24 74 69)(33 42 81)(34 43 82)(35 44 83)(36 45 84)(37 46 85)(38 47 86)(39 48 87)(40 41 88)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 89)(25 85)(26 86)(27 87)(28 88)(29 81)(30 82)(31 83)(32 84)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)(57 80)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)
(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 79 5 75)(2 78 6 74)(3 77 7 73)(4 76 8 80)(9 85 13 81)(10 84 14 88)(11 83 15 87)(12 82 16 86)(17 54 21 50)(18 53 22 49)(19 52 23 56)(20 51 24 55)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(33 64 37 60)(34 63 38 59)(35 62 39 58)(36 61 40 57)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C ··· 12I 12J 12K 12L ··· 12Q 12R 12S 12T 12U 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 ··· 12 12 12 12 ··· 12 12 12 12 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 3 3 1 1 2 2 2 2 4 4 6 12 12 1 1 2 2 2 3 3 3 3 2 2 6 6 2 2 4 ··· 4 6 6 8 ··· 8 12 12 12 12 2 2 2 2 4 ··· 4 6 6 6 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 Q16 C3×S3 C3×D4 C3×D4 S3×C6 S3×C6 C3×Q16 S3×D4 S3×Q16 C3×S3×D4 C3×S3×Q16 kernel C3×S3×Q16 S3×C24 C3×Dic12 C3×C3⋊Q16 C32×Q16 C3×S3×Q8 S3×Q16 S3×C8 Dic12 C3⋊Q16 C3×Q16 S3×Q8 C3×Q16 C3×Dic3 S3×C6 C24 C3×Q8 C3×S3 Q16 Dic3 D6 C8 Q8 S3 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 1 1 2 4 2 2 2 2 4 8 1 2 2 4

Matrix representation of C3×S3×Q16 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 4 4 4 6 1 5 6 3 5 5 6 1 3 4 5 4
,
 4 2 5 3 6 2 1 4 0 6 5 4 5 0 3 3
,
 6 3 0 2 1 6 1 1 2 2 2 3 3 4 2 6
,
 3 6 6 1 6 3 6 6 6 6 5 0 2 5 0 3
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[3,6,6,2,6,3,6,5,6,6,5,0,1,6,0,3] >;

C3×S3×Q16 in GAP, Magma, Sage, TeX

C_3\times S_3\times Q_{16}
% in TeX

G:=Group("C3xS3xQ16");
// GroupNames label

G:=SmallGroup(288,688);
// by ID

G=gap.SmallGroup(288,688);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,303,268,1271,648,102,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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