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## G = S3×C6×Q8order 288 = 25·32

### Direct product of C6, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×C6×Q8
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — S3×C2×C12 — S3×C6×Q8
 Lower central C3 — C6 — S3×C6×Q8
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for S3×C6×Q8
G = < a,b,c,d,e | a6=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 586 in 331 conjugacy classes, 194 normal (20 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, Q8, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C2×Q8, C2×Q8, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C22×Q8, C3×Dic3, C3×C12, S3×C6, C62, C2×Dic6, S3×C2×C4, S3×Q8, C22×C12, C6×Q8, C6×Q8, C3×Dic6, S3×C12, C6×Dic3, C6×C12, Q8×C32, S3×C2×C6, C2×S3×Q8, Q8×C2×C6, C6×Dic6, S3×C2×C12, C3×S3×Q8, Q8×C3×C6, S3×C6×Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C24, C3×S3, C3×Q8, C22×S3, C22×C6, C22×Q8, S3×C6, S3×Q8, C6×Q8, S3×C23, C23×C6, S3×C2×C6, C2×S3×Q8, Q8×C2×C6, C3×S3×Q8, S3×C22×C6, S3×C6×Q8

Smallest permutation representation of S3×C6×Q8
On 96 points
Generators in S96
(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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)(49 51 53)(50 52 54)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 78)(8 73)(9 74)(10 75)(11 76)(12 77)(13 64)(14 65)(15 66)(16 61)(17 62)(18 63)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(31 82)(32 83)(33 84)(34 79)(35 80)(36 81)(37 88)(38 89)(39 90)(40 85)(41 86)(42 87)(43 94)(44 95)(45 96)(46 91)(47 92)(48 93)
(1 22 41 35)(2 23 42 36)(3 24 37 31)(4 19 38 32)(5 20 39 33)(6 21 40 34)(7 48 53 17)(8 43 54 18)(9 44 49 13)(10 45 50 14)(11 46 51 15)(12 47 52 16)(25 63 73 94)(26 64 74 95)(27 65 75 96)(28 66 76 91)(29 61 77 92)(30 62 78 93)(55 67 86 80)(56 68 87 81)(57 69 88 82)(58 70 89 83)(59 71 90 84)(60 72 85 79)
(1 47 41 16)(2 48 42 17)(3 43 37 18)(4 44 38 13)(5 45 39 14)(6 46 40 15)(7 36 53 23)(8 31 54 24)(9 32 49 19)(10 33 50 20)(11 34 51 21)(12 35 52 22)(25 69 73 82)(26 70 74 83)(27 71 75 84)(28 72 76 79)(29 67 77 80)(30 68 78 81)(55 92 86 61)(56 93 87 62)(57 94 88 63)(58 95 89 64)(59 96 90 65)(60 91 85 66)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A ··· 4F 4G ··· 4L 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12L 12M ··· 12AD 12AE ··· 12AP order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 3 3 3 3 1 1 2 2 2 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2 4 ··· 4 6 ··· 6

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 Q8 D6 D6 C3×S3 C3×Q8 S3×C6 S3×C6 S3×Q8 C3×S3×Q8 kernel S3×C6×Q8 C6×Dic6 S3×C2×C12 C3×S3×Q8 Q8×C3×C6 C2×S3×Q8 C2×Dic6 S3×C2×C4 S3×Q8 C6×Q8 C6×Q8 S3×C6 C2×C12 C3×Q8 C2×Q8 D6 C2×C4 Q8 C6 C2 # reps 1 3 3 8 1 2 6 6 16 2 1 4 3 4 2 8 6 8 2 4

Matrix representation of S3×C6×Q8 in GL4(𝔽13) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 3 0 0 0 0 1 0 0 0 0 1
,
 0 12 0 0 12 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 12 0 0 0 0 9 10 0 0 10 4
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,9,10,0,0,10,4] >;

S3×C6×Q8 in GAP, Magma, Sage, TeX

S_3\times C_6\times Q_8
% in TeX

G:=Group("S3xC6xQ8");
// GroupNames label

G:=SmallGroup(288,995);
// by ID

G=gap.SmallGroup(288,995);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,268,409,192,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^4=1,e^2=d^2,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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