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G = C22×C4×C3⋊S3order 288 = 25·32

Direct product of C22×C4 and C3⋊S3

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

Aliases: C22×C4×C3⋊S3, C62.275C23, (C2×C12)⋊30D6, C128(C22×S3), C6221(C2×C4), C326(C23×C4), (C3×C12)⋊10C23, (C6×C12)⋊36C22, (C22×C12)⋊14S3, C6.55(S3×C23), (C3×C6).54C24, C3⋊Dic39C23, (C22×C6).165D6, (C2×C62).121C22, C63(S3×C2×C4), (C2×C6×C12)⋊17C2, C33(S3×C22×C4), (C2×C6)⋊15(C4×S3), (C3×C6)⋊6(C22×C4), C2.1(C23×C3⋊S3), (C23×C3⋊S3).8C2, C23.39(C2×C3⋊S3), (C2×C3⋊S3).59C23, (C2×C6).284(C22×S3), (C2×C3⋊Dic3)⋊30C22, (C22×C3⋊Dic3)⋊17C2, C22.29(C22×C3⋊S3), (C22×C3⋊S3).113C22, SmallGroup(288,1004)

Series: Derived Chief Lower central Upper central

C1C32 — C22×C4×C3⋊S3
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3C23×C3⋊S3 — C22×C4×C3⋊S3
C32 — C22×C4×C3⋊S3
C1C22×C4

Generators and relations for C22×C4×C3⋊S3
 G = < a,b,c,d,e,f | a2=b2=c4=d3=e3=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, fdf=d-1, fef=e-1 >

Subgroups: 2164 in 708 conjugacy classes, 253 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C4 [×4], C22 [×7], C22 [×28], S3 [×32], C6 [×28], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], C32, Dic3 [×16], C12 [×16], D6 [×112], C2×C6 [×28], C22×C4, C22×C4 [×13], C24, C3⋊S3 [×8], C3×C6, C3×C6 [×6], C4×S3 [×64], C2×Dic3 [×24], C2×C12 [×24], C22×S3 [×56], C22×C6 [×4], C23×C4, C3⋊Dic3 [×4], C3×C12 [×4], C2×C3⋊S3 [×28], C62 [×7], S3×C2×C4 [×48], C22×Dic3 [×4], C22×C12 [×4], S3×C23 [×4], C4×C3⋊S3 [×16], C2×C3⋊Dic3 [×6], C6×C12 [×6], C22×C3⋊S3 [×14], C2×C62, S3×C22×C4 [×4], C2×C4×C3⋊S3 [×12], C22×C3⋊Dic3, C2×C6×C12, C23×C3⋊S3, C22×C4×C3⋊S3
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3 [×4], C2×C4 [×28], C23 [×15], D6 [×28], C22×C4 [×14], C24, C3⋊S3, C4×S3 [×16], C22×S3 [×28], C23×C4, C2×C3⋊S3 [×7], S3×C2×C4 [×24], S3×C23 [×4], C4×C3⋊S3 [×4], C22×C3⋊S3 [×7], S3×C22×C4 [×4], C2×C4×C3⋊S3 [×6], C23×C3⋊S3, C22×C4×C3⋊S3

Smallest permutation representation of C22×C4×C3⋊S3
On 144 points
Generators in S144
(1 27)(2 28)(3 25)(4 26)(5 133)(6 134)(7 135)(8 136)(9 51)(10 52)(11 49)(12 50)(13 137)(14 138)(15 139)(16 140)(17 59)(18 60)(19 57)(20 58)(21 126)(22 127)(23 128)(24 125)(29 120)(30 117)(31 118)(32 119)(33 113)(34 114)(35 115)(36 116)(37 144)(38 141)(39 142)(40 143)(41 79)(42 80)(43 77)(44 78)(45 83)(46 84)(47 81)(48 82)(53 91)(54 92)(55 89)(56 90)(61 99)(62 100)(63 97)(64 98)(65 103)(66 104)(67 101)(68 102)(69 107)(70 108)(71 105)(72 106)(73 111)(74 112)(75 109)(76 110)(85 123)(86 124)(87 121)(88 122)(93 131)(94 132)(95 129)(96 130)
(1 76)(2 73)(3 74)(4 75)(5 61)(6 62)(7 63)(8 64)(9 123)(10 124)(11 121)(12 122)(13 65)(14 66)(15 67)(16 68)(17 131)(18 132)(19 129)(20 130)(21 54)(22 55)(23 56)(24 53)(25 112)(26 109)(27 110)(28 111)(29 48)(30 45)(31 46)(32 47)(33 41)(34 42)(35 43)(36 44)(37 72)(38 69)(39 70)(40 71)(49 87)(50 88)(51 85)(52 86)(57 95)(58 96)(59 93)(60 94)(77 115)(78 116)(79 113)(80 114)(81 119)(82 120)(83 117)(84 118)(89 127)(90 128)(91 125)(92 126)(97 135)(98 136)(99 133)(100 134)(101 139)(102 140)(103 137)(104 138)(105 143)(106 144)(107 141)(108 142)
(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 21 31)(2 22 32)(3 23 29)(4 24 30)(5 58 16)(6 59 13)(7 60 14)(8 57 15)(9 115 141)(10 116 142)(11 113 143)(12 114 144)(17 137 134)(18 138 135)(19 139 136)(20 140 133)(25 128 120)(26 125 117)(27 126 118)(28 127 119)(33 40 49)(34 37 50)(35 38 51)(36 39 52)(41 71 87)(42 72 88)(43 69 85)(44 70 86)(45 75 53)(46 76 54)(47 73 55)(48 74 56)(61 96 68)(62 93 65)(63 94 66)(64 95 67)(77 107 123)(78 108 124)(79 105 121)(80 106 122)(81 111 89)(82 112 90)(83 109 91)(84 110 92)(97 132 104)(98 129 101)(99 130 102)(100 131 103)
(1 35 7)(2 36 8)(3 33 5)(4 34 6)(9 138 118)(10 139 119)(11 140 120)(12 137 117)(13 30 50)(14 31 51)(15 32 52)(16 29 49)(17 125 144)(18 126 141)(19 127 142)(20 128 143)(21 38 60)(22 39 57)(23 40 58)(24 37 59)(25 113 133)(26 114 134)(27 115 135)(28 116 136)(41 61 74)(42 62 75)(43 63 76)(44 64 73)(45 88 65)(46 85 66)(47 86 67)(48 87 68)(53 72 93)(54 69 94)(55 70 95)(56 71 96)(77 97 110)(78 98 111)(79 99 112)(80 100 109)(81 124 101)(82 121 102)(83 122 103)(84 123 104)(89 108 129)(90 105 130)(91 106 131)(92 107 132)
(1 74)(2 75)(3 76)(4 73)(5 43)(6 44)(7 41)(8 42)(9 130)(10 131)(11 132)(12 129)(13 70)(14 71)(15 72)(16 69)(17 124)(18 121)(19 122)(20 123)(21 48)(22 45)(23 46)(24 47)(25 110)(26 111)(27 112)(28 109)(29 54)(30 55)(31 56)(32 53)(33 63)(34 64)(35 61)(36 62)(37 67)(38 68)(39 65)(40 66)(49 94)(50 95)(51 96)(52 93)(57 88)(58 85)(59 86)(60 87)(77 133)(78 134)(79 135)(80 136)(81 125)(82 126)(83 127)(84 128)(89 117)(90 118)(91 119)(92 120)(97 113)(98 114)(99 115)(100 116)(101 144)(102 141)(103 142)(104 143)(105 138)(106 139)(107 140)(108 137)

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

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

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

96 conjugacy classes

class 1 2A···2G2H···2O3A3B3C3D4A···4H4I···4P6A···6AB12A···12AF
order12···22···233334···44···46···612···12
size11···19···922221···19···92···22···2

96 irreducible representations

dim1111112222
type++++++++
imageC1C2C2C2C2C4S3D6D6C4×S3
kernelC22×C4×C3⋊S3C2×C4×C3⋊S3C22×C3⋊Dic3C2×C6×C12C23×C3⋊S3C22×C3⋊S3C22×C12C2×C12C22×C6C2×C6
# reps11211116424432

Matrix representation of C22×C4×C3⋊S3 in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
00000100
00000010
00000001
,
120000000
012000000
001200000
000120000
000012000
000001200
00000010
00000001
,
50000000
05000000
001200000
000120000
000012000
000001200
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
000000121
000000120
,
1212000000
10000000
0012120000
00100000
0000121200
00001000
000000012
000000112
,
10000000
1212000000
001200000
00110000
000012000
00001100
00000001
00000010

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C22×C4×C3⋊S3 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times C_3\rtimes S_3
% in TeX

G:=Group("C2^2xC4xC3:S3");
// GroupNames label

G:=SmallGroup(288,1004);
// by ID

G=gap.SmallGroup(288,1004);
# by ID

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

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^3=e^3=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,f*d*f=d^-1,f*e*f=e^-1>;
// generators/relations

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