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## G = S3×C3⋊Q16order 288 = 25·32

### Direct product of S3 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — S3×C3⋊Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — S3×C3⋊Q16
 Lower central C32 — C3×C6 — C3×C12 — S3×C3⋊Q16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for S3×C3⋊Q16
G = < a,b,c,d,e | a3=b2=c3=d8=1, e2=d4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 434 in 129 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×2], C6 [×2], C6 [×3], C8 [×2], C2×C4 [×3], Q8, Q8 [×5], C32, Dic3, Dic3 [×5], C12 [×2], C12 [×7], D6, C2×C6, C2×C8, Q16 [×4], C2×Q8 [×2], C3×S3 [×2], C3×C6, C3⋊C8, C3⋊C8 [×3], C24, Dic6 [×2], Dic6 [×6], C4×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×2], C3×Q8 [×2], C3×Q8 [×4], C2×Q16, C3×Dic3, C3×Dic3 [×2], C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C8, Dic12, C2×C3⋊C8, C3⋊Q16, C3⋊Q16 [×6], C3×Q16, C2×Dic6, S3×Q8, S3×Q8, C6×Q8, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6 [×2], C3×Dic6, S3×C12, S3×C12, C324Q8, Q8×C32, S3×Q16, C2×C3⋊Q16, S3×C3⋊C8, C322Q16, C323Q16, C3×C3⋊Q16, C327Q16, S3×Dic6, C3×S3×Q8, S3×C3⋊Q16
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], Q16 [×2], C2×D4, C3⋊D4 [×2], C22×S3 [×2], C2×Q16, S32, C3⋊Q16 [×2], S3×D4, C2×C3⋊D4, C2×S32, S3×Q16, C2×C3⋊Q16, S3×C3⋊D4, S3×C3⋊Q16

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

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 24A 24B order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 3 3 2 2 4 2 4 6 12 12 36 2 2 4 6 6 6 6 18 18 4 4 4 4 8 8 8 8 12 12 12 24 12 12

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + + + + + + + + - + - + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 Q16 C3⋊D4 C3⋊D4 S32 C3⋊Q16 S3×D4 C2×S32 S3×Q16 S3×C3⋊D4 S3×C3⋊Q16 kernel S3×C3⋊Q16 S3×C3⋊C8 C32⋊2Q16 C32⋊3Q16 C3×C3⋊Q16 C32⋊7Q16 S3×Dic6 C3×S3×Q8 C3⋊Q16 S3×Q8 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 C3×Q8 C3×S3 Dic3 D6 Q8 S3 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 4 2 2 1 2 1 1 2 2 1

Matrix representation of S3×C3⋊Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 57 57 0 0 0 0 16 57 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 62 43 0 0 0 0 43 11 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,16,0,0,0,0,57,57,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[62,43,0,0,0,0,43,11,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;

S3×C3⋊Q16 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC3:Q16");
// GroupNames label

G:=SmallGroup(288,590);
// by ID

G=gap.SmallGroup(288,590);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,100,346,185,80,1356,9414]);
// Polycyclic

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

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