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G = C24×C3⋊S3  order 288 = 25·32

Direct product of C24 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group, rational

Aliases: C24×C3⋊S3, C32⋊3C25, C62⋊11C23, C6⋊2(S3×C23), (C3×C6)⋊3C24, C3⋊2(S3×C24), (C23×C6)⋊11S3, (C22×C6)⋊17D6, (C22×C62)⋊8C2, (C2×C62)⋊18C22, (C2×C6)⋊12(C22×S3), SmallGroup(288,1044)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C24×C3⋊S3
Chief series
C1 — C3 — C32 — C3⋊S3 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3 — C24×C3⋊S3
Lower central
C32 — C24×C3⋊S3
Upper central
C1 — C24

Generators and relations for C24×C3⋊S3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e3=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, geg=e-1, gfg=f-1 >

Subgroups: 7156 in 2244 conjugacy classes, 709 normal (5 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, C32, D6, C2×C6, C24, C24, C3⋊S3, C3×C6, C22×S3, C22×C6, C25, C2×C3⋊S3, C62, S3×C23, C23×C6, C22×C3⋊S3, C2×C62, S3×C24, C23×C3⋊S3, C22×C62, C24×C3⋊S3
Quotients: C1, C2, C22, S3, C23, D6, C24, C3⋊S3, C22×S3, C25, C2×C3⋊S3, S3×C23, C22×C3⋊S3, S3×C24, C23×C3⋊S3, C24×C3⋊S3

Smallest permutation representation of C24×C3⋊S3
►On 144 points
Generators in S144
(1 137)(2 138)(3 136)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 127)(11 128)(12 129)(13 130)(14 131)(15 132)(16 133)(17 134)(18 135)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)

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

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

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

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

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

96 conjugacy classes

class 1 2A···2O2P···2AE3A3B3C3D6A···6BH
order12···22···233336···6
size11···19···922222···2

96 irreducible representations

dim11122
type+++++
imageC1C2C2S3D6
kernelC24×C3⋊S3C23×C3⋊S3C22×C62C23×C6C22×C6
# reps1301460

Matrix representation of C24×C3⋊S3 ►in GL6(ℤ)

100000
0-10000
00-1000
000-100
0000-10
00000-1
,
100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
010000
001000
000100
0000-10
00000-1
,
100000
010000
000100
00-1-100
000001
0000-1-1
,
100000
010000
001000
000100
0000-1-1
000010
,
100000
0-10000
001000
00-1-100
000011
00000-1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,-1] >;
 

C24×C3⋊S3 in GAP, Magma, Sage, TeX 

C_2^4\times C_3\rtimes S_3
 
% in TeX
 
G:=Group("C2^4xC3:S3");
 
// GroupNames label
 
G:=SmallGroup(288,1044);
 
// by ID
 
G=gap.SmallGroup(288,1044);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^3=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,g*e*g=e^-1,g*f*g=f^-1>;
 
// generators/relations
 

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