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## G = C2×S3×Dic6order 288 = 25·32

### Direct product of C2, S3 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×S3×Dic6
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — S3×Dic3 — C2×S3×Dic3 — C2×S3×Dic6
 Lower central C32 — C3×C6 — C2×S3×Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×Dic6
G = < a,b,c,d,e | a2=b3=c2=d12=1, e2=d6, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 994 in 331 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×10], C22, C22 [×6], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C2×C4, C2×C4 [×17], Q8 [×16], C23, C32, Dic3 [×6], Dic3 [×12], C12 [×4], C12 [×8], D6 [×6], C2×C6 [×2], C2×C6 [×7], C22×C4 [×3], C2×Q8 [×12], C3×S3 [×4], C3×C6, C3×C6 [×2], Dic6 [×4], Dic6 [×28], C4×S3 [×4], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×14], C2×C12 [×2], C2×C12 [×8], C3×Q8 [×4], C22×S3, C22×C6, C22×Q8, C3×Dic3 [×6], C3⋊Dic3 [×4], C3×C12 [×2], S3×C6 [×6], C62, C2×Dic6, C2×Dic6 [×15], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×8], C22×Dic3 [×2], C22×C12, C6×Q8, S3×Dic3 [×8], C322Q8 [×8], C3×Dic6 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C324Q8 [×4], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C22×Dic6, C2×S3×Q8, S3×Dic6 [×8], C2×S3×Dic3 [×2], C2×C322Q8 [×2], C6×Dic6, S3×C2×C12, C2×C324Q8, C2×S3×Dic6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], Q8 [×4], C23 [×15], D6 [×14], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×14], C22×Q8, S32, C2×Dic6 [×6], S3×Q8 [×2], S3×C23 [×2], C2×S32 [×3], C22×Dic6, C2×S3×Q8, S3×Dic6 [×2], C22×S32, C2×S3×Dic6

Smallest permutation representation of C2×S3×Dic6
On 96 points
Generators in S96
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 49)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 88)(26 89)(27 90)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 85)(35 86)(36 87)(61 83)(62 84)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 61)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 49)(46 50)(47 51)(48 52)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 85)(82 86)(83 87)(84 88)
(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 93 7 87)(2 92 8 86)(3 91 9 85)(4 90 10 96)(5 89 11 95)(6 88 12 94)(13 74 19 80)(14 73 20 79)(15 84 21 78)(16 83 22 77)(17 82 23 76)(18 81 24 75)(25 49 31 55)(26 60 32 54)(27 59 33 53)(28 58 34 52)(29 57 35 51)(30 56 36 50)(37 64 43 70)(38 63 44 69)(39 62 45 68)(40 61 46 67)(41 72 47 66)(42 71 48 65)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P 12Q 12R order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 12 12 size 1 1 1 1 3 3 3 3 2 2 4 2 2 6 ··· 6 18 18 18 18 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 6 6 6 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + + - + - + + - image C1 C2 C2 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D6 Dic6 S32 S3×Q8 C2×S32 C2×S32 S3×Dic6 kernel C2×S3×Dic6 S3×Dic6 C2×S3×Dic3 C2×C32⋊2Q8 C6×Dic6 S3×C2×C12 C2×C32⋊4Q8 C2×Dic6 S3×C2×C4 S3×C6 Dic6 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 1 4 4 4 3 2 1 8 1 2 2 1 4

Matrix representation of C2×S3×Dic6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 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 1 0 0 0 0 12 12 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 12 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 11 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 12 12 0 0 0 0 1 0
,
 6 5 0 0 0 0 3 7 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[6,3,0,0,0,0,5,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;

C2×S3×Dic6 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_6
% in TeX

G:=Group("C2xS3xDic6");
// GroupNames label

G:=SmallGroup(288,942);
// by ID

G=gap.SmallGroup(288,942);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^12=1,e^2=d^6,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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