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G = C3×D48order 288 = 25·32

Direct product of C3 and D48

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D48, C481C6, C483S3, D241C6, C324D16, C24.79D6, C6.19D24, C12.62D12, C161(C3×S3), (C3×C48)⋊2C2, C31(C3×D16), C6.1(C3×D8), (C3×D24)⋊4C2, C8.13(S3×C6), C2.3(C3×D24), C4.1(C3×D12), (C3×C6).18D8, C24.16(C2×C6), C12.24(C3×D4), (C3×C12).126D4, (C3×C24).54C22, SmallGroup(288,233)

Series: Derived Chief Lower central Upper central

C1C24 — C3×D48
C1C3C6C12C24C3×C24C3×D24 — C3×D48
C3C6C12C24 — C3×D48
C1C6C12C24C48

Generators and relations for C3×D48
 G = < a,b,c | a3=b48=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 278 in 61 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, D4 [×2], C32, C12 [×2], C12, D6 [×2], C2×C6 [×2], C16, D8 [×2], C3×S3 [×2], C3×C6, C24 [×2], C24, D12 [×2], C3×D4 [×2], D16, C3×C12, S3×C6 [×2], C48 [×2], C48, D24 [×2], C3×D8 [×2], C3×C24, C3×D12 [×2], D48, C3×D16, C3×C48, C3×D24 [×2], C3×D48
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, D8, C3×S3, D12, C3×D4, D16, S3×C6, D24, C3×D8, C3×D12, D48, C3×D16, C3×D24, C3×D48

Smallest permutation representation of C3×D48
On 96 points
Generators in S96
(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 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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 49)(26 96)(27 95)(28 94)(29 93)(30 92)(31 91)(32 90)(33 89)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)

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

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

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

81 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I8A8B12A···12H16A16B16C16D24A···24P48A···48AF
order12223333346666666668812···121616161624···2448···48
size1124241122221122224242424222···222222···22···2

81 irreducible representations

dim1111112222222222222222
type+++++++++++
imageC1C2C2C3C6C6S3D4D6D8C3×S3D12C3×D4D16S3×C6D24C3×D8C3×D12D48C3×D16C3×D24C3×D48
kernelC3×D48C3×C48C3×D24D48C48D24C48C3×C12C24C3×C6C16C12C12C32C8C6C6C4C3C3C2C1
# reps11222411122224244488816

Matrix representation of C3×D48 in GL2(𝔽97) generated by

350
035
,
490
02
,
02
490
G:=sub<GL(2,GF(97))| [35,0,0,35],[49,0,0,2],[0,49,2,0] >;

C3×D48 in GAP, Magma, Sage, TeX

C_3\times D_{48}
% in TeX

G:=Group("C3xD48");
// GroupNames label

G:=SmallGroup(288,233);
// by ID

G=gap.SmallGroup(288,233);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,260,1011,192,2524,102,9414]);
// Polycyclic

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

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×
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