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G = C3×C48order 144 = 24·32

Abelian group of type [3,48]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C48, SmallGroup(144,30)

Series: Derived Chief Lower central Upper central

C1 — C3×C48
C1C2C4C8C24C3×C24 — C3×C48
C1 — C3×C48
C1 — C3×C48

Generators and relations for C3×C48
 G = < a,b | a3=b48=1, ab=ba >


Smallest permutation representation of C3×C48
Regular action on 144 points
Generators in S144
(1 69 117)(2 70 118)(3 71 119)(4 72 120)(5 73 121)(6 74 122)(7 75 123)(8 76 124)(9 77 125)(10 78 126)(11 79 127)(12 80 128)(13 81 129)(14 82 130)(15 83 131)(16 84 132)(17 85 133)(18 86 134)(19 87 135)(20 88 136)(21 89 137)(22 90 138)(23 91 139)(24 92 140)(25 93 141)(26 94 142)(27 95 143)(28 96 144)(29 49 97)(30 50 98)(31 51 99)(32 52 100)(33 53 101)(34 54 102)(35 55 103)(36 56 104)(37 57 105)(38 58 106)(39 59 107)(40 60 108)(41 61 109)(42 62 110)(43 63 111)(44 64 112)(45 65 113)(46 66 114)(47 67 115)(48 68 116)
(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)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,69,117)(2,70,118)(3,71,119)(4,72,120)(5,73,121)(6,74,122)(7,75,123)(8,76,124)(9,77,125)(10,78,126)(11,79,127)(12,80,128)(13,81,129)(14,82,130)(15,83,131)(16,84,132)(17,85,133)(18,86,134)(19,87,135)(20,88,136)(21,89,137)(22,90,138)(23,91,139)(24,92,140)(25,93,141)(26,94,142)(27,95,143)(28,96,144)(29,49,97)(30,50,98)(31,51,99)(32,52,100)(33,53,101)(34,54,102)(35,55,103)(36,56,104)(37,57,105)(38,58,106)(39,59,107)(40,60,108)(41,61,109)(42,62,110)(43,63,111)(44,64,112)(45,65,113)(46,66,114)(47,67,115)(48,68,116), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)>;

G:=Group( (1,69,117)(2,70,118)(3,71,119)(4,72,120)(5,73,121)(6,74,122)(7,75,123)(8,76,124)(9,77,125)(10,78,126)(11,79,127)(12,80,128)(13,81,129)(14,82,130)(15,83,131)(16,84,132)(17,85,133)(18,86,134)(19,87,135)(20,88,136)(21,89,137)(22,90,138)(23,91,139)(24,92,140)(25,93,141)(26,94,142)(27,95,143)(28,96,144)(29,49,97)(30,50,98)(31,51,99)(32,52,100)(33,53,101)(34,54,102)(35,55,103)(36,56,104)(37,57,105)(38,58,106)(39,59,107)(40,60,108)(41,61,109)(42,62,110)(43,63,111)(44,64,112)(45,65,113)(46,66,114)(47,67,115)(48,68,116), (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)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144) );

G=PermutationGroup([(1,69,117),(2,70,118),(3,71,119),(4,72,120),(5,73,121),(6,74,122),(7,75,123),(8,76,124),(9,77,125),(10,78,126),(11,79,127),(12,80,128),(13,81,129),(14,82,130),(15,83,131),(16,84,132),(17,85,133),(18,86,134),(19,87,135),(20,88,136),(21,89,137),(22,90,138),(23,91,139),(24,92,140),(25,93,141),(26,94,142),(27,95,143),(28,96,144),(29,49,97),(30,50,98),(31,51,99),(32,52,100),(33,53,101),(34,54,102),(35,55,103),(36,56,104),(37,57,105),(38,58,106),(39,59,107),(40,60,108),(41,61,109),(42,62,110),(43,63,111),(44,64,112),(45,65,113),(46,66,114),(47,67,115),(48,68,116)], [(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),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)])

C3×C48 is a maximal subgroup of   C48.S3  C48⋊S3  C325D16  C6.D24  C325Q32

144 conjugacy classes

class 1  2 3A···3H4A4B6A···6H8A8B8C8D12A···12P16A···16H24A···24AF48A···48BL
order123···3446···6888812···1216···1624···2448···48
size111···1111···111111···11···11···11···1

144 irreducible representations

dim1111111111
type++
imageC1C2C3C4C6C8C12C16C24C48
kernelC3×C48C3×C24C48C3×C12C24C3×C6C12C32C6C3
# reps1182841683264

Matrix representation of C3×C48 in GL2(𝔽97) generated by

610
01
,
360
048
G:=sub<GL(2,GF(97))| [61,0,0,1],[36,0,0,48] >;

C3×C48 in GAP, Magma, Sage, TeX

C_3\times C_{48}
% in TeX

G:=Group("C3xC48");
// GroupNames label

G:=SmallGroup(144,30);
// by ID

G=gap.SmallGroup(144,30);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-2,-2,108,69,88]);
// Polycyclic

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

Export

Subgroup lattice of C3×C48 in TeX

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