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G = C24×D9order 288 = 25·32

Direct product of C24 and D9

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C24×D9, C9⋊C25, C18⋊C24, C3.(S3×C24), (C2×C18)⋊4C23, (C23×C18)⋊5C2, C6.53(S3×C23), (C23×C6).13S3, (C22×C18)⋊8C22, (C22×C6).163D6, (C2×C6).282(C22×S3), SmallGroup(288,839)

Series: Derived Chief Lower central Upper central

C1C9 — C24×D9
C1C3C9D9D18C22×D9C23×D9 — C24×D9
C9 — C24×D9
C1C24

Generators and relations for C24×D9
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e9=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 4192 in 1122 conjugacy classes, 508 normal (7 characteristic)
C1, C2 [×15], C2 [×16], C3, C22 [×35], C22 [×120], S3 [×16], C6 [×15], C23 [×15], C23 [×140], C9, D6 [×120], C2×C6 [×35], C24, C24 [×30], D9 [×16], C18 [×15], C22×S3 [×140], C22×C6 [×15], C25, D18 [×120], C2×C18 [×35], S3×C23 [×30], C23×C6, C22×D9 [×140], C22×C18 [×15], S3×C24, C23×D9 [×30], C23×C18, C24×D9
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], D9, C22×S3 [×35], C25, D18 [×15], S3×C23 [×15], C22×D9 [×35], S3×C24, C23×D9 [×15], C24×D9

Smallest permutation representation of C24×D9
On 144 points
Generators in S144
(1 86)(2 87)(3 88)(4 89)(5 90)(6 82)(7 83)(8 84)(9 85)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 55)(7 56)(8 57)(9 58)(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 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 127)(119 128)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(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 67)(2 66)(3 65)(4 64)(5 72)(6 71)(7 70)(8 69)(9 68)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 56)(17 55)(18 63)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 54)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 45)(73 143)(74 142)(75 141)(76 140)(77 139)(78 138)(79 137)(80 136)(81 144)(82 134)(83 133)(84 132)(85 131)(86 130)(87 129)(88 128)(89 127)(90 135)(91 125)(92 124)(93 123)(94 122)(95 121)(96 120)(97 119)(98 118)(99 126)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(108 117)

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

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

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

96 conjugacy classes

class 1 2A···2O2P···2AE 3 6A···6O9A9B9C18A···18AS
order12···22···236···699918···18
size11···19···922···22222···2

96 irreducible representations

dim1112222
type+++++++
imageC1C2C2S3D6D9D18
kernelC24×D9C23×D9C23×C18C23×C6C22×C6C24C23
# reps1301115345

Matrix representation of C24×D9 in GL5(𝔽19)

180000
018000
00100
00010
00001
,
10000
018000
001800
00010
00001
,
10000
018000
001800
000180
000018
,
10000
018000
00100
000180
000018
,
10000
01000
00100
000714
00052
,
180000
018000
00100
0001417
000125

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,5,0,0,0,14,2],[18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,14,12,0,0,0,17,5] >;

C24×D9 in GAP, Magma, Sage, TeX

C_2^4\times D_9
% in TeX

G:=Group("C2^4xD9");
// GroupNames label

G:=SmallGroup(288,839);
// by ID

G=gap.SmallGroup(288,839);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^9=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

׿
×
𝔽