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G = C3×C42.S3order 288 = 25·32

Direct product of C3 and C42.S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×C42.S3, C122.11C2, C3⋊C83C12, (C4×C12).3S3, C6.2(C4×C12), C4.19(S3×C12), (C2×C12).8C12, (C4×C12).13C6, (C6×C12).15C4, C42.1(C3×S3), C12.110(C4×S3), C12.24(C2×C12), C325(C8⋊C4), (C2×C12).451D6, (C3×C6).12C42, C62.93(C2×C4), (C2×C12).9Dic3, C2.3(Dic3×C12), C6.18(C4×Dic3), C6.4(C3×M4(2)), (C3×C6).18M4(2), C22.8(C6×Dic3), (C6×C12).329C22, C6.11(C4.Dic3), (C3×C3⋊C8)⋊9C4, (C2×C3⋊C8).7C6, C31(C3×C8⋊C4), (C6×C3⋊C8).21C2, (C2×C4).88(S3×C6), (C2×C6).35(C2×C12), (C2×C4).2(C3×Dic3), (C3×C12).104(C2×C4), (C2×C12).118(C2×C6), C2.1(C3×C4.Dic3), (C2×C6).57(C2×Dic3), SmallGroup(288,237)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C42.S3
C1C3C6C2×C6C2×C12C6×C12C6×C3⋊C8 — C3×C42.S3
C3C6 — C3×C42.S3
C1C2×C12C4×C12

Generators and relations for C3×C42.S3
 G = < a,b,c,d | a3=b6=d4=1, c4=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c >

Subgroups: 146 in 95 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, C12 [×4], C12 [×10], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×3], C8⋊C4, C3×C12 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×C12 [×2], C4×C12, C2×C24 [×2], C3×C3⋊C8 [×4], C6×C12, C6×C12 [×2], C42.S3, C3×C8⋊C4, C6×C3⋊C8 [×2], C122, C3×C42.S3
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, M4(2) [×2], C3×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×3], C8⋊C4, C3×Dic3 [×2], S3×C6, C4.Dic3 [×2], C4×Dic3, C4×C12, C3×M4(2) [×2], S3×C12 [×2], C6×Dic3, C42.S3, C3×C8⋊C4, C3×C4.Dic3 [×2], Dic3×C12, C3×C42.S3

Smallest permutation representation of C3×C42.S3
On 96 points
Generators in S96
(1 69 27)(2 70 28)(3 71 29)(4 72 30)(5 65 31)(6 66 32)(7 67 25)(8 68 26)(9 63 83)(10 64 84)(11 57 85)(12 58 86)(13 59 87)(14 60 88)(15 61 81)(16 62 82)(17 40 90)(18 33 91)(19 34 92)(20 35 93)(21 36 94)(22 37 95)(23 38 96)(24 39 89)(41 53 79)(42 54 80)(43 55 73)(44 56 74)(45 49 75)(46 50 76)(47 51 77)(48 52 78)
(1 31 69 5 27 65)(2 66 28 6 70 32)(3 25 71 7 29 67)(4 68 30 8 72 26)(9 87 63 13 83 59)(10 60 84 14 64 88)(11 81 57 15 85 61)(12 62 86 16 58 82)(17 36 90 21 40 94)(18 95 33 22 91 37)(19 38 92 23 34 96)(20 89 35 24 93 39)(41 75 53 45 79 49)(42 50 80 46 54 76)(43 77 55 47 73 51)(44 52 74 48 56 78)
(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 83 22 51)(2 88 23 56)(3 85 24 53)(4 82 17 50)(5 87 18 55)(6 84 19 52)(7 81 20 49)(8 86 21 54)(9 37 77 69)(10 34 78 66)(11 39 79 71)(12 36 80 68)(13 33 73 65)(14 38 74 70)(15 35 75 67)(16 40 76 72)(25 61 93 45)(26 58 94 42)(27 63 95 47)(28 60 96 44)(29 57 89 41)(30 62 90 46)(31 59 91 43)(32 64 92 48)

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

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

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

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6O8A···8H12A···12H12I···12AZ24A···24P
order122233333444444446···66···68···812···1212···1224···24
size111111222111122221···12···26···61···12···26···6

108 irreducible representations

dim1111111111222222222222
type++++-+
imageC1C2C2C3C4C4C6C6C12C12S3Dic3D6M4(2)C3×S3C4×S3C3×Dic3S3×C6C4.Dic3C3×M4(2)S3×C12C3×C4.Dic3
kernelC3×C42.S3C6×C3⋊C8C122C42.S3C3×C3⋊C8C6×C12C2×C3⋊C8C4×C12C3⋊C8C2×C12C4×C12C2×C12C2×C12C3×C6C42C12C2×C4C2×C4C6C6C4C2
# reps121284421681214244288816

Matrix representation of C3×C42.S3 in GL3(𝔽73) generated by

800
0640
0064
,
100
06555
009
,
100
05418
02219
,
2700
0115
0072
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[1,0,0,0,65,0,0,55,9],[1,0,0,0,54,22,0,18,19],[27,0,0,0,1,0,0,15,72] >;

C3×C42.S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2.S_3
% in TeX

G:=Group("C3xC4^2.S3");
// GroupNames label

G:=SmallGroup(288,237);
// by ID

G=gap.SmallGroup(288,237);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,136,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=d^4=1,c^4=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

׿
×
𝔽