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G = S3×C48order 288 = 25·32

Direct product of C48 and S3

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

Aliases: S3×C48, C485C6, D6.2C24, C24.97D6, Dic3.2C24, C3⋊C166C6, C31(C2×C48), (C3×C48)⋊7C2, C3⋊C8.3C12, (S3×C8).3C6, (S3×C6).4C8, C8.19(S3×C6), C2.1(S3×C24), C6.22(S3×C8), C6.1(C2×C24), C327(C2×C16), (S3×C24).6C2, (S3×C12).9C4, (C4×S3).4C12, C4.16(S3×C12), C24.32(C2×C6), C12.107(C4×S3), C12.21(C2×C12), (C3×Dic3).4C8, (C3×C24).64C22, (C3×C3⋊C8).6C4, (C3×C3⋊C16)⋊13C2, (C3×C6).27(C2×C8), (C3×C12).101(C2×C4), SmallGroup(288,231)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C48
C1C3C6C12C24C3×C24S3×C24 — S3×C48
C3 — S3×C48
C1C48

Generators and relations for S3×C48
 G = < a,b,c | a48=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 102 in 61 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4, C22, S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16, C16, C2×C8, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C2×C16, C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48 [×2], C48 [×2], S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, S3×C16, C2×C48, C3×C3⋊C16, C3×C48, S3×C24, S3×C48
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C8 [×2], C2×C4, C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3, C24 [×2], C4×S3, C2×C12, C2×C16, S3×C6, C48 [×2], S3×C8, C2×C24, S3×C12, S3×C16, C2×C48, S3×C24, S3×C48

Smallest permutation representation of S3×C48
On 96 points
Generators in S96
(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 17 33)(2 18 34)(3 19 35)(4 20 36)(5 21 37)(6 22 38)(7 23 39)(8 24 40)(9 25 41)(10 26 42)(11 27 43)(12 28 44)(13 29 45)(14 30 46)(15 31 47)(16 32 48)(49 81 65)(50 82 66)(51 83 67)(52 84 68)(53 85 69)(54 86 70)(55 87 71)(56 88 72)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)

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

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

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

144 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I8A8B8C8D8E8F8G8H12A12B12C12D12E···12J12K12L12M12N16A···16H16I···16P24A···24H24I···24T24U···24AB48A···48P48Q···48AN48AO···48BD
order1222333334444666666666888888881212121212···121212121216···1616···1624···2424···2424···2448···4848···4848···48
size11331122211331122233331111333311112···233331···13···31···12···23···31···12···23···3

144 irreducible representations

dim1111111111111111112222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C8C12C12C16C24C24C48S3D6C3×S3C4×S3S3×C6S3×C8S3×C12S3×C16S3×C24S3×C48
kernelS3×C48C3×C3⋊C16C3×C48S3×C24S3×C16C3×C3⋊C8S3×C12C3⋊C16C48S3×C8C3×Dic3S3×C6C3⋊C8C4×S3C3×S3Dic3D6S3C48C24C16C12C8C6C4C3C2C1
# reps1111222222444416883211222448816

Matrix representation of S3×C48 in GL3(𝔽97) generated by

6600
0880
0088
,
100
0350
06161
,
9600
09660
001
G:=sub<GL(3,GF(97))| [66,0,0,0,88,0,0,0,88],[1,0,0,0,35,61,0,0,61],[96,0,0,0,96,0,0,60,1] >;

S3×C48 in GAP, Magma, Sage, TeX

S_3\times C_{48}
% in TeX

G:=Group("S3xC48");
// GroupNames label

G:=SmallGroup(288,231);
// by ID

G=gap.SmallGroup(288,231);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,92,80,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^48=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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