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## G = C4×S3×Dic3order 288 = 25·32

### Direct product of C4, S3 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C4×S3×Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C4×S3×Dic3
 Lower central C32 — C4×S3×Dic3
 Upper central C1 — C2×C4

Generators and relations for C4×S3×Dic3
G = < a,b,c,d,e | a4=b3=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 618 in 231 conjugacy classes, 104 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], C23, C32, Dic3 [×6], Dic3 [×12], C12 [×4], C12 [×8], D6 [×6], C2×C6 [×2], C2×C6 [×7], C42 [×4], C22×C4 [×3], C3×S3 [×4], C3×C6, C3×C6 [×2], C4×S3 [×4], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×14], C2×C12 [×2], C2×C12 [×8], C22×S3, C22×C6, C2×C42, C3×Dic3 [×6], C3⋊Dic3 [×4], C3×C12 [×2], S3×C6 [×6], C62, C4×Dic3, C4×Dic3 [×7], C4×C12, S3×C2×C4, S3×C2×C4 [×2], C22×Dic3 [×2], C22×C12, S3×Dic3 [×8], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, S3×C42, C2×C4×Dic3, Dic32 [×2], Dic3×C12, C4×C3⋊Dic3, C2×S3×Dic3 [×2], S3×C2×C12, C4×S3×Dic3
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3 [×2], C2×C4 [×18], C23, Dic3 [×4], D6 [×6], C42 [×4], C22×C4 [×3], C4×S3 [×10], C2×Dic3 [×6], C22×S3 [×2], C2×C42, S32, C4×Dic3 [×4], S3×C2×C4 [×5], C22×Dic3, S3×Dic3 [×2], C2×S32, S3×C42, C2×C4×Dic3, C4×S32 [×2], C2×S3×Dic3, C4×S3×Dic3

Smallest permutation representation of C4×S3×Dic3
On 96 points
Generators in S96
(1 22 16 27)(2 23 17 28)(3 24 18 29)(4 19 13 30)(5 20 14 25)(6 21 15 26)(7 88 95 83)(8 89 96 84)(9 90 91 79)(10 85 92 80)(11 86 93 81)(12 87 94 82)(31 53 42 43)(32 54 37 44)(33 49 38 45)(34 50 39 46)(35 51 40 47)(36 52 41 48)(55 78 65 67)(56 73 66 68)(57 74 61 69)(58 75 62 70)(59 76 63 71)(60 77 64 72)
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(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 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 83 81)(80 84 82)(85 89 87)(86 90 88)(91 95 93)(92 96 94)
(1 65)(2 66)(3 61)(4 62)(5 63)(6 64)(7 54)(8 49)(9 50)(10 51)(11 52)(12 53)(13 58)(14 59)(15 60)(16 55)(17 56)(18 57)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 76)(26 77)(27 78)(28 73)(29 74)(30 75)(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 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 51 4 54)(2 50 5 53)(3 49 6 52)(7 65 10 62)(8 64 11 61)(9 63 12 66)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 37 22 40)(20 42 23 39)(21 41 24 38)(25 31 28 34)(26 36 29 33)(27 35 30 32)(55 92 58 95)(56 91 59 94)(57 96 60 93)(67 85 70 88)(68 90 71 87)(69 89 72 86)(73 79 76 82)(74 84 77 81)(75 83 78 80)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E ··· 4P 4Q ··· 4X 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 12A ··· 12H 12I 12J 12K 12L 12M ··· 12X 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 size 1 1 1 1 3 3 3 3 2 2 4 1 1 1 1 3 ··· 3 9 ··· 9 2 ··· 2 4 4 4 6 6 6 6 2 ··· 2 4 4 4 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 S3 S3 Dic3 D6 D6 D6 C4×S3 C4×S3 C4×S3 S32 S3×Dic3 C2×S32 C4×S32 kernel C4×S3×Dic3 Dic32 Dic3×C12 C4×C3⋊Dic3 C2×S3×Dic3 S3×C2×C12 S3×Dic3 S3×C12 C4×Dic3 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 Dic3 C12 D6 C2×C4 C4 C22 C2 # reps 1 2 1 1 2 1 16 8 1 1 4 3 2 1 8 4 8 1 2 1 4

Matrix representation of C4×S3×Dic3 in GL4(𝔽13) generated by

 5 0 0 0 0 5 0 0 0 0 8 0 0 0 0 8
,
 0 1 0 0 12 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 1 1 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 1
,
 1 0 0 0 0 1 0 0 0 0 5 8 0 0 0 8
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8],[0,12,0,0,1,12,0,0,0,0,1,0,0,0,0,1],[12,1,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,5,0,0,0,8,8] >;

C4×S3×Dic3 in GAP, Magma, Sage, TeX

C_4\times S_3\times {\rm Dic}_3
% in TeX

G:=Group("C4xS3xDic3");
// GroupNames label

G:=SmallGroup(288,523);
// by ID

G=gap.SmallGroup(288,523);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,100,1356,9414]);
// Polycyclic

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