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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, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, Q16, C2×Q8, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C8, Dic12, C2×C3⋊C8, C3⋊Q16, C3⋊Q16, C3×Q16, C2×Dic6, S3×Q8, S3×Q8, C6×Q8, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6, 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, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C2×Q16, S32, C3⋊Q16, 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 48 87)(10 41 88)(11 42 81)(12 43 82)(13 44 83)(14 45 84)(15 46 85)(16 47 86)(17 72 73)(18 65 74)(19 66 75)(20 67 76)(21 68 77)(22 69 78)(23 70 79)(24 71 80)(33 58 96)(34 59 89)(35 60 90)(36 61 91)(37 62 92)(38 63 93)(39 64 94)(40 57 95)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)
(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 48 87)(10 88 41)(11 42 81)(12 82 43)(13 44 83)(14 84 45)(15 46 85)(16 86 47)(17 73 72)(18 65 74)(19 75 66)(20 67 76)(21 77 68)(22 69 78)(23 79 70)(24 71 80)(33 58 96)(34 89 59)(35 60 90)(36 91 61)(37 62 92)(38 93 63)(39 64 94)(40 95 57)
(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 19 5 23)(2 18 6 22)(3 17 7 21)(4 24 8 20)(9 58 13 62)(10 57 14 61)(11 64 15 60)(12 63 16 59)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(49 80 53 76)(50 79 54 75)(51 78 55 74)(52 77 56 73)

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

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

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

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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