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G = S3xC4xC12order 288 = 25·32

Direct product of C4xC12 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3xC4xC12, C122:13C2, C62.162C23, (C4xC12):14C6, C12:5(C2xC12), D6.6(C2xC12), C32:5(C2xC42), (C4xDic3):17C6, Dic3:5(C2xC12), (C2xC12).458D6, C6.2(C22xC12), (Dic3xC12):35C2, (C6xC12).344C22, (C6xDic3).165C22, C3:1(C2xC4xC12), C2.1(S3xC2xC12), (S3xC2xC4).11C6, C6.101(S3xC2xC4), (C3xC12):19(C2xC4), C22.9(S3xC2xC6), (S3xC2xC12).24C2, (C2xC4).96(S3xC6), (S3xC6).25(C2xC4), (C2xC12).126(C2xC6), (C3xDic3):18(C2xC4), (S3xC2xC6).113C22, (C2xC6).17(C22xC6), (C3xC6).73(C22xC4), (C22xS3).33(C2xC6), (C2xC6).295(C22xS3), (C2xDic3).46(C2xC6), SmallGroup(288,642)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC4xC12
C1C3C6C2xC6C62S3xC2xC6S3xC2xC12 — S3xC4xC12
C3 — S3xC4xC12
C1C4xC12

Generators and relations for S3xC4xC12
 G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 402 in 231 conjugacy classes, 138 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C42, C22xC4, C3xS3, C3xC6, C4xS3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, C2xC42, C3xDic3, C3xC12, S3xC6, C62, C4xDic3, C4xC12, C4xC12, S3xC2xC4, C22xC12, S3xC12, C6xDic3, C6xC12, S3xC2xC6, S3xC42, C2xC4xC12, Dic3xC12, C122, S3xC2xC12, S3xC4xC12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C23, C12, D6, C2xC6, C42, C22xC4, C3xS3, C4xS3, C2xC12, C22xS3, C22xC6, C2xC42, S3xC6, C4xC12, S3xC2xC4, C22xC12, S3xC12, S3xC2xC6, S3xC42, C2xC4xC12, S3xC2xC12, S3xC4xC12

Smallest permutation representation of S3xC4xC12
On 96 points
Generators in S96
(1 41 35 56)(2 42 36 57)(3 43 25 58)(4 44 26 59)(5 45 27 60)(6 46 28 49)(7 47 29 50)(8 48 30 51)(9 37 31 52)(10 38 32 53)(11 39 33 54)(12 40 34 55)(13 63 96 78)(14 64 85 79)(15 65 86 80)(16 66 87 81)(17 67 88 82)(18 68 89 83)(19 69 90 84)(20 70 91 73)(21 71 92 74)(22 72 93 75)(23 61 94 76)(24 62 95 77)
(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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 49)(24 50)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)

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

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

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

144 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D3E4A···4L4M···4X6A···6F6G···6O6P···6W12A···12X12Y···12BH12BI···12CF
order12222222333334···44···46···66···66···612···1212···1212···12
size11113333112221···13···31···12···23···31···12···23···3

144 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12S3D6C3xS3C4xS3S3xC6S3xC12
kernelS3xC4xC12Dic3xC12C122S3xC2xC12S3xC42S3xC12C4xDic3C4xC12S3xC2xC4C4xS3C4xC12C2xC12C42C12C2xC4C4
# reps13132246264813212624

Matrix representation of S3xC4xC12 in GL3(F13) generated by

100
080
008
,
700
070
007
,
100
090
063
,
1200
0121
001
G:=sub<GL(3,GF(13))| [1,0,0,0,8,0,0,0,8],[7,0,0,0,7,0,0,0,7],[1,0,0,0,9,6,0,0,3],[12,0,0,0,12,0,0,1,1] >;

S3xC4xC12 in GAP, Magma, Sage, TeX

S_3\times C_4\times C_{12}
% in TeX

G:=Group("S3xC4xC12");
// GroupNames label

G:=SmallGroup(288,642);
// by ID

G=gap.SmallGroup(288,642);
# by ID

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

G:=Group<a,b,c,d|a^4=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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