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G = C22×C32⋊7D4  order 288 = 25·32

Direct product of C22 and C32⋊7D4

direct product, metabelian, supersoluble, monomial

Aliases: C22×C32⋊7D4, C62⋊26D4, C62⋊9C23, C24⋊6(C3⋊S3), (C23×C6)⋊10S3, (C22×C6)⋊16D6, C6.68(S3×C23), (C3×C6).67C24, C3⋊Dic3⋊8C23, (C22×C62)⋊6C2, C32⋊14(C22×D4), (C2×C62)⋊16C22, C6⋊4(C2×C3⋊D4), (C3×C6)⋊13(C2×D4), C23⋊5(C2×C3⋊S3), (C2×C3⋊S3)⋊8C23, (C23×C3⋊S3)⋊8C2, C3⋊4(C22×C3⋊D4), (C2×C6)⋊18(C3⋊D4), (C2×C6)⋊11(C22×S3), C2.15(C23×C3⋊S3), C22⋊3(C22×C3⋊S3), (C22×C3⋊S3)⋊17C22, (C2×C3⋊Dic3)⋊29C22, (C22×C3⋊Dic3)⋊16C2, SmallGroup(288,1017)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C22×C32⋊7D4
Chief series
C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C23×C3⋊S3 — C22×C32⋊7D4
Lower central
C32 — C3×C6 — C22×C32⋊7D4

Subgroups: 2404 in 708 conjugacy classes, 213 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C22 [×11], C22 [×28], S3 [×16], C6 [×28], C6 [×16], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], C32, Dic3 [×16], D6 [×64], C2×C6 [×44], C2×C6 [×48], C22×C4, C2×D4 [×12], C24, C24, C3⋊S3 [×4], C3×C6, C3×C6 [×6], C3×C6 [×4], C2×Dic3 [×24], C3⋊D4 [×64], C22×S3 [×40], C22×C6 [×28], C22×C6 [×16], C22×D4, C3⋊Dic3 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62 [×11], C62 [×12], C22×Dic3 [×4], C2×C3⋊D4 [×48], S3×C23 [×4], C23×C6 [×4], C2×C3⋊Dic3 [×6], C32⋊7D4 [×16], C22×C3⋊S3 [×6], C22×C3⋊S3 [×4], C2×C62, C2×C62 [×6], C2×C62 [×4], C22×C3⋊D4 [×4], C22×C3⋊Dic3, C2×C32⋊7D4 [×12], C23×C3⋊S3, C22×C62, C22×C32⋊7D4

Quotients:
C1, C2 [×15], C22 [×35], S3 [×4], D4 [×4], C23 [×15], D6 [×28], C2×D4 [×6], C24, C3⋊S3, C3⋊D4 [×16], C22×S3 [×28], C22×D4, C2×C3⋊S3 [×7], C2×C3⋊D4 [×24], S3×C23 [×4], C32⋊7D4 [×4], C22×C3⋊S3 [×7], C22×C3⋊D4 [×4], C2×C32⋊7D4 [×6], C23×C3⋊S3, C22×C32⋊7D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c3=d3=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=fcf=c-1, ede-1=fdf=d-1, fef=e-1 >

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽13)

120000
012000
001200
00010
00001
,
120000
01000
00100
000120
000012
,
10000
0121200
01000
00001
0001212
,
10000
00100
0121200
00001
0001212
,
120000
012000
01100
000119
000112
,
120000
01000
0121200
000120
00011

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,1,12],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,1,12],[12,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,11,11,0,0,0,9,2],[12,0,0,0,0,0,1,12,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,1] >;
 

84 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O3A3B3C3D4A4B4C4D6A···6BH
order12···222222222333344446···6
size11···12222181818182222181818182···2

84 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2S3D4D6C3⋊D4
kernelC22×C32⋊7D4C22×C3⋊Dic3C2×C32⋊7D4C23×C3⋊S3C22×C62C23×C6C62C22×C6C2×C6
# reps111211442832

In GAP, Magma, Sage, TeX 

C_2^2\times C_3^2\rtimes_7D_4
 
% in TeX
 
G:=Group("C2^2xC3^2:7D4");
 
// GroupNames label
 
G:=SmallGroup(288,1017);
 
// by ID
 
G=gap.SmallGroup(288,1017);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,675,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^4=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,e*c*e^-1=f*c*f=c^-1,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;
 
// generators/relations
 

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