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G = C3×D48  order 288 = 25·32

Direct product of C3 and D48

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D48, C48⋊1C6, C48⋊3S3, D24⋊1C6, C32⋊4D16, C24.79D6, C6.19D24, C12.62D12, C16⋊1(C3×S3), (C3×C48)⋊2C2, C3⋊1(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

Derived series
C1 — C24 — C3×D48
Chief series
C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×D24 — C3×D48
Lower central
C3 — C6 — C12 — C24 — C3×D48
Upper central
C1 — C6 — C12 — C24 — C48

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, C3, C3, C4, C22, S3, C6, C6, C8, D4, C32, C12, C12, D6, C2×C6, C16, D8, C3×S3, C3×C6, C24, C24, D12, C3×D4, D16, C3×C12, S3×C6, C48, C48, D24, C3×D8, C3×C24, C3×D12, D48, C3×D16, C3×C48, C3×D24, C3×D48
Quotients: C1, C2, C3, C22, S3, C6, 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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
 
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)>;
 
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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57) );
 
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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,96),(10,95),(11,94),(12,93),(13,92),(14,91),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,84),(22,83),(23,82),(24,81),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)]])
 

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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