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## G = C3×S3×C4⋊C4order 288 = 25·32

### Direct product of C3, S3 and C4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×S3×C4⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — S3×C2×C12 — C3×S3×C4⋊C4
 Lower central C3 — C6 — C3×S3×C4⋊C4
 Upper central C1 — C2×C6 — C3×C4⋊C4

Generators and relations for C3×S3×C4⋊C4
G = < a,b,c,d,e | a3=b3=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 402 in 197 conjugacy classes, 98 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C32, Dic3 [×2], Dic3 [×2], C12 [×4], C12 [×12], D6 [×6], C2×C6 [×2], C2×C6 [×7], C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×14], C22×S3, C22×C6, C2×C4⋊C4, C3×Dic3 [×2], C3×Dic3 [×2], C3×C12 [×2], C3×C12 [×2], S3×C6 [×6], C62, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], S3×C2×C4, S3×C2×C4 [×2], C22×C12 [×3], S3×C12 [×4], S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C6×C12, C6×C12 [×2], S3×C2×C6, S3×C4⋊C4, C6×C4⋊C4, C3×Dic3⋊C4 [×2], C3×C4⋊Dic3, C32×C4⋊C4, S3×C2×C12, S3×C2×C12 [×2], C3×S3×C4⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3×S3, C4×S3 [×2], C2×C12 [×6], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C2×C4⋊C4, S3×C6 [×3], C3×C4⋊C4 [×4], S3×C2×C4, S3×D4, S3×Q8, C22×C12, C6×D4, C6×Q8, S3×C12 [×2], S3×C2×C6, S3×C4⋊C4, C6×C4⋊C4, S3×C2×C12, C3×S3×D4, C3×S3×Q8, C3×S3×C4⋊C4

Smallest permutation representation of C3×S3×C4⋊C4
On 96 points
Generators in S96
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 60 76)(6 57 73)(7 58 74)(8 59 75)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)(29 34 83)(30 35 84)(31 36 81)(32 33 82)(49 54 89)(50 55 90)(51 56 91)(52 53 92)(61 66 79)(62 67 80)(63 68 77)(64 65 78)(69 85 94)(70 86 95)(71 87 96)(72 88 93)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 60 76)(6 57 73)(7 58 74)(8 59 75)(9 25 42)(10 26 43)(11 27 44)(12 28 41)(21 37 46)(22 38 47)(23 39 48)(24 40 45)(29 83 34)(30 84 35)(31 81 36)(32 82 33)(49 89 54)(50 90 55)(51 91 56)(52 92 53)(61 79 66)(62 80 67)(63 77 68)(64 78 65)(69 94 85)(70 95 86)(71 96 87)(72 93 88)
(1 78)(2 79)(3 80)(4 77)(5 83)(6 84)(7 81)(8 82)(9 50)(10 51)(11 52)(12 49)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 55)(26 56)(27 53)(28 54)(29 60)(30 57)(31 58)(32 59)(33 75)(34 76)(35 73)(36 74)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 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 47 43 8)(2 46 44 7)(3 45 41 6)(4 48 42 5)(9 60 15 23)(10 59 16 22)(11 58 13 21)(12 57 14 24)(17 38 26 75)(18 37 27 74)(19 40 28 73)(20 39 25 76)(29 63 71 50)(30 62 72 49)(31 61 69 52)(32 64 70 51)(33 65 86 56)(34 68 87 55)(35 67 88 54)(36 66 85 53)(77 96 90 83)(78 95 91 82)(79 94 92 81)(80 93 89 84)

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

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

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

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 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C12 S3 D4 Q8 D6 C3×S3 C4×S3 C3×D4 C3×Q8 S3×C6 S3×C12 S3×D4 S3×Q8 C3×S3×D4 C3×S3×Q8 kernel C3×S3×C4⋊C4 C3×Dic3⋊C4 C3×C4⋊Dic3 C32×C4⋊C4 S3×C2×C12 S3×C4⋊C4 S3×C12 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 S3×C2×C4 C4×S3 C3×C4⋊C4 S3×C6 S3×C6 C2×C12 C4⋊C4 C12 D6 D6 C2×C4 C4 C6 C6 C2 C2 # reps 1 2 1 1 3 2 8 4 2 2 6 16 1 2 2 3 2 4 4 4 6 8 1 1 2 2

Matrix representation of C3×S3×C4⋊C4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 3 3 0 0 0 0 1 0 0 0 0 1
,
 12 2 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 12 2 0 0 12 1
,
 5 0 0 0 0 5 0 0 0 0 1 5 0 0 10 12
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,3,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,2,1],[5,0,0,0,0,5,0,0,0,0,1,10,0,0,5,12] >;

C3×S3×C4⋊C4 in GAP, Magma, Sage, TeX

C_3\times S_3\times C_4\rtimes C_4
% in TeX

G:=Group("C3xS3xC4:C4");
// GroupNames label

G:=SmallGroup(288,662);
// by ID

G=gap.SmallGroup(288,662);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,555,142,9414]);
// Polycyclic

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