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G = C48⋊S3order 288 = 25·32

6th semidirect product of C48 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C486S3, C24.92D6, C328M5(2), (C3×C48)⋊9C2, C163(C3⋊S3), C6.13(S3×C8), C12.76(C4×S3), C32(D6.C8), C3⋊Dic3.5C8, C24.S38C2, C324C8.6C4, (C3×C24).69C22, C2.3(C8×C3⋊S3), (C4×C3⋊S3).9C4, (C8×C3⋊S3).5C2, (C2×C3⋊S3).5C8, C8.19(C2×C3⋊S3), C4.17(C4×C3⋊S3), (C3×C6).33(C2×C8), (C3×C12).108(C2×C4), SmallGroup(288,273)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C48⋊S3
C1C3C32C3×C6C3×C12C3×C24C8×C3⋊S3 — C48⋊S3
C32C3×C6 — C48⋊S3
C1C8C16

Generators and relations for C48⋊S3
 G = < a,b,c | a48=b3=c2=1, ab=ba, cac=a41, cbc=b-1 >

Subgroups: 228 in 78 conjugacy classes, 37 normal (17 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, C8, C2×C4, C32, Dic3 [×4], C12 [×4], D6 [×4], C16, C16, C2×C8, C3⋊S3, C3×C6, C3⋊C8 [×4], C24 [×4], C4×S3 [×4], M5(2), C3⋊Dic3, C3×C12, C2×C3⋊S3, C3⋊C16 [×4], C48 [×4], S3×C8 [×4], C324C8, C3×C24, C4×C3⋊S3, D6.C8 [×4], C24.S3, C3×C48, C8×C3⋊S3, C48⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D6 [×4], C2×C8, C3⋊S3, C4×S3 [×4], M5(2), C2×C3⋊S3, S3×C8 [×4], C4×C3⋊S3, D6.C8 [×4], C8×C3⋊S3, C48⋊S3

Smallest permutation representation of C48⋊S3
On 144 points
Generators in S144
(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)
(1 120 87)(2 121 88)(3 122 89)(4 123 90)(5 124 91)(6 125 92)(7 126 93)(8 127 94)(9 128 95)(10 129 96)(11 130 49)(12 131 50)(13 132 51)(14 133 52)(15 134 53)(16 135 54)(17 136 55)(18 137 56)(19 138 57)(20 139 58)(21 140 59)(22 141 60)(23 142 61)(24 143 62)(25 144 63)(26 97 64)(27 98 65)(28 99 66)(29 100 67)(30 101 68)(31 102 69)(32 103 70)(33 104 71)(34 105 72)(35 106 73)(36 107 74)(37 108 75)(38 109 76)(39 110 77)(40 111 78)(41 112 79)(42 113 80)(43 114 81)(44 115 82)(45 116 83)(46 117 84)(47 118 85)(48 119 86)
(2 42)(3 35)(4 28)(5 21)(6 14)(8 48)(9 41)(10 34)(11 27)(12 20)(15 47)(16 40)(17 33)(18 26)(22 46)(23 39)(24 32)(29 45)(30 38)(36 44)(49 98)(50 139)(51 132)(52 125)(53 118)(54 111)(55 104)(56 97)(57 138)(58 131)(59 124)(60 117)(61 110)(62 103)(63 144)(64 137)(65 130)(66 123)(67 116)(68 109)(69 102)(70 143)(71 136)(72 129)(73 122)(74 115)(75 108)(76 101)(77 142)(78 135)(79 128)(80 121)(81 114)(82 107)(83 100)(84 141)(85 134)(86 127)(87 120)(88 113)(89 106)(90 99)(91 140)(92 133)(93 126)(94 119)(95 112)(96 105)

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

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

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

84 conjugacy classes

class 1 2A2B3A3B3C3D4A4B4C6A6B6C6D8A8B8C8D8E8F12A···12H16A16B16C16D16E16F16G16H24A···24P48A···48AF
order1223333444666688888812···12161616161616161624···2448···48
size1118222211182222111118182···22222181818182···22···2

84 irreducible representations

dim11111111222222
type++++++
imageC1C2C2C2C4C4C8C8S3D6C4×S3M5(2)S3×C8D6.C8
kernelC48⋊S3C24.S3C3×C48C8×C3⋊S3C324C8C4×C3⋊S3C3⋊Dic3C2×C3⋊S3C48C24C12C32C6C3
# reps1111224444841632

Matrix representation of C48⋊S3 in GL6(𝔽97)

47470000
5000000
0050300
00834700
0000750
0000075
,
96960000
100000
001000
000100
00009696
000010
,
100000
96960000
001000
00969600
000001
000010

G:=sub<GL(6,GF(97))| [47,50,0,0,0,0,47,0,0,0,0,0,0,0,50,83,0,0,0,0,3,47,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[96,1,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,1,0,0,0,0,96,0],[1,96,0,0,0,0,0,96,0,0,0,0,0,0,1,96,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C48⋊S3 in GAP, Magma, Sage, TeX

C_{48}\rtimes S_3
% in TeX

G:=Group("C48:S3");
// GroupNames label

G:=SmallGroup(288,273);
// by ID

G=gap.SmallGroup(288,273);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^48=b^3=c^2=1,a*b=b*a,c*a*c=a^41,c*b*c=b^-1>;
// generators/relations

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