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G = C3×D48order 288 = 25·32

Direct product of C3 and D48

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 2A 2B 2C 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 8A 8B 12A ··· 12H 16A 16B 16C 16D 24A ··· 24P 48A ··· 48AF order 1 2 2 2 3 3 3 3 3 4 6 6 6 6 6 6 6 6 6 8 8 12 ··· 12 16 16 16 16 24 ··· 24 48 ··· 48 size 1 1 24 24 1 1 2 2 2 2 1 1 2 2 2 24 24 24 24 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 D8 C3×S3 D12 C3×D4 D16 S3×C6 D24 C3×D8 C3×D12 D48 C3×D16 C3×D24 C3×D48 kernel C3×D48 C3×C48 C3×D24 D48 C48 D24 C48 C3×C12 C24 C3×C6 C16 C12 C12 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 4 2 4 4 4 8 8 8 16

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

 35 0 0 35
,
 49 0 0 2
,
 0 2 49 0
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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