direct product, metabelian, supersoluble, monomial, A-group
Aliases: C22×C4×C3⋊S3, C62.275C23, (C2×C12)⋊30D6, C12⋊8(C22×S3), C62⋊21(C2×C4), C32⋊6(C23×C4), (C3×C12)⋊10C23, (C6×C12)⋊36C22, (C22×C12)⋊14S3, C6.55(S3×C23), (C3×C6).54C24, C3⋊Dic3⋊9C23, (C22×C6).165D6, (C2×C62).121C22, C6⋊3(S3×C2×C4), (C2×C6×C12)⋊17C2, C3⋊3(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
| C32 — C22×C4×C3⋊S3 |
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, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C22×C4, C24, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C23×C4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C62, S3×C22×C4, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C6×C12, C23×C3⋊S3, C22×C4×C3⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C3⋊S3, C4×S3, C22×S3, C23×C4, C2×C3⋊S3, S3×C2×C4, S3×C23, C4×C3⋊S3, C22×C3⋊S3, S3×C22×C4, C2×C4×C3⋊S3, C23×C3⋊S3, C22×C4×C3⋊S3
►On 144 points
Generators in S144
(1 9)(2 10)(3 11)(4 12)(5 116)(6 113)(7 114)(8 115)(13 136)(14 133)(15 134)(16 135)(17 50)(18 51)(19 52)(20 49)(21 140)(22 137)(23 138)(24 139)(25 58)(26 59)(27 60)(28 57)(29 120)(30 117)(31 118)(32 119)(33 143)(34 144)(35 141)(36 142)(37 125)(38 126)(39 127)(40 128)(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 44)(6 41)(7 42)(8 43)(9 110)(10 111)(11 112)(12 109)(13 64)(14 61)(15 62)(16 63)(17 122)(18 123)(19 124)(20 121)(21 68)(22 65)(23 66)(24 67)(25 130)(26 131)(27 132)(28 129)(29 48)(30 45)(31 46)(32 47)(33 71)(34 72)(35 69)(36 70)(37 53)(38 54)(39 55)(40 56)(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 16 31)(2 13 32)(3 14 29)(4 15 30)(5 36 57)(6 33 58)(7 34 59)(8 35 60)(9 135 118)(10 136 119)(11 133 120)(12 134 117)(17 137 125)(18 138 126)(19 139 127)(20 140 128)(21 40 49)(22 37 50)(23 38 51)(24 39 52)(25 113 143)(26 114 144)(27 115 141)(28 116 142)(41 71 96)(42 72 93)(43 69 94)(44 70 95)(45 75 62)(46 76 63)(47 73 64)(48 74 61)(53 88 65)(54 85 66)(55 86 67)(56 87 68)(77 107 132)(78 108 129)(79 105 130)(80 106 131)(81 111 98)(82 112 99)(83 109 100)(84 110 97)(89 124 101)(90 121 102)(91 122 103)(92 123 104)
(1 8 38)(2 5 39)(3 6 40)(4 7 37)(9 115 126)(10 116 127)(11 113 128)(12 114 125)(13 36 52)(14 33 49)(15 34 50)(16 35 51)(17 134 144)(18 135 141)(19 136 142)(20 133 143)(21 29 58)(22 30 59)(23 31 60)(24 32 57)(25 140 120)(26 137 117)(27 138 118)(28 139 119)(41 56 74)(42 53 75)(43 54 76)(44 55 73)(45 93 65)(46 94 66)(47 95 67)(48 96 68)(61 71 87)(62 72 88)(63 69 85)(64 70 86)(77 92 110)(78 89 111)(79 90 112)(80 91 109)(81 129 101)(82 130 102)(83 131 103)(84 132 104)(97 107 123)(98 108 124)(99 105 121)(100 106 122)
(1 74)(2 75)(3 76)(4 73)(5 53)(6 54)(7 55)(8 56)(9 112)(10 109)(11 110)(12 111)(13 45)(14 46)(15 47)(16 48)(17 129)(18 130)(19 131)(20 132)(21 69)(22 70)(23 71)(24 72)(25 123)(26 124)(27 121)(28 122)(29 63)(30 64)(31 61)(32 62)(33 66)(34 67)(35 68)(36 65)(37 44)(38 41)(39 42)(40 43)(49 94)(50 95)(51 96)(52 93)(57 88)(58 85)(59 86)(60 87)(77 128)(78 125)(79 126)(80 127)(81 134)(82 135)(83 136)(84 133)(89 114)(90 115)(91 116)(92 113)(97 120)(98 117)(99 118)(100 119)(101 144)(102 141)(103 142)(104 143)(105 138)(106 139)(107 140)(108 137)
G:=sub<Sym(144)| (1,9)(2,10)(3,11)(4,12)(5,116)(6,113)(7,114)(8,115)(13,136)(14,133)(15,134)(16,135)(17,50)(18,51)(19,52)(20,49)(21,140)(22,137)(23,138)(24,139)(25,58)(26,59)(27,60)(28,57)(29,120)(30,117)(31,118)(32,119)(33,143)(34,144)(35,141)(36,142)(37,125)(38,126)(39,127)(40,128)(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)(5,44)(6,41)(7,42)(8,43)(9,110)(10,111)(11,112)(12,109)(13,64)(14,61)(15,62)(16,63)(17,122)(18,123)(19,124)(20,121)(21,68)(22,65)(23,66)(24,67)(25,130)(26,131)(27,132)(28,129)(29,48)(30,45)(31,46)(32,47)(33,71)(34,72)(35,69)(36,70)(37,53)(38,54)(39,55)(40,56)(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), (1,16,31)(2,13,32)(3,14,29)(4,15,30)(5,36,57)(6,33,58)(7,34,59)(8,35,60)(9,135,118)(10,136,119)(11,133,120)(12,134,117)(17,137,125)(18,138,126)(19,139,127)(20,140,128)(21,40,49)(22,37,50)(23,38,51)(24,39,52)(25,113,143)(26,114,144)(27,115,141)(28,116,142)(41,71,96)(42,72,93)(43,69,94)(44,70,95)(45,75,62)(46,76,63)(47,73,64)(48,74,61)(53,88,65)(54,85,66)(55,86,67)(56,87,68)(77,107,132)(78,108,129)(79,105,130)(80,106,131)(81,111,98)(82,112,99)(83,109,100)(84,110,97)(89,124,101)(90,121,102)(91,122,103)(92,123,104), (1,8,38)(2,5,39)(3,6,40)(4,7,37)(9,115,126)(10,116,127)(11,113,128)(12,114,125)(13,36,52)(14,33,49)(15,34,50)(16,35,51)(17,134,144)(18,135,141)(19,136,142)(20,133,143)(21,29,58)(22,30,59)(23,31,60)(24,32,57)(25,140,120)(26,137,117)(27,138,118)(28,139,119)(41,56,74)(42,53,75)(43,54,76)(44,55,73)(45,93,65)(46,94,66)(47,95,67)(48,96,68)(61,71,87)(62,72,88)(63,69,85)(64,70,86)(77,92,110)(78,89,111)(79,90,112)(80,91,109)(81,129,101)(82,130,102)(83,131,103)(84,132,104)(97,107,123)(98,108,124)(99,105,121)(100,106,122), (1,74)(2,75)(3,76)(4,73)(5,53)(6,54)(7,55)(8,56)(9,112)(10,109)(11,110)(12,111)(13,45)(14,46)(15,47)(16,48)(17,129)(18,130)(19,131)(20,132)(21,69)(22,70)(23,71)(24,72)(25,123)(26,124)(27,121)(28,122)(29,63)(30,64)(31,61)(32,62)(33,66)(34,67)(35,68)(36,65)(37,44)(38,41)(39,42)(40,43)(49,94)(50,95)(51,96)(52,93)(57,88)(58,85)(59,86)(60,87)(77,128)(78,125)(79,126)(80,127)(81,134)(82,135)(83,136)(84,133)(89,114)(90,115)(91,116)(92,113)(97,120)(98,117)(99,118)(100,119)(101,144)(102,141)(103,142)(104,143)(105,138)(106,139)(107,140)(108,137)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,116)(6,113)(7,114)(8,115)(13,136)(14,133)(15,134)(16,135)(17,50)(18,51)(19,52)(20,49)(21,140)(22,137)(23,138)(24,139)(25,58)(26,59)(27,60)(28,57)(29,120)(30,117)(31,118)(32,119)(33,143)(34,144)(35,141)(36,142)(37,125)(38,126)(39,127)(40,128)(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)(5,44)(6,41)(7,42)(8,43)(9,110)(10,111)(11,112)(12,109)(13,64)(14,61)(15,62)(16,63)(17,122)(18,123)(19,124)(20,121)(21,68)(22,65)(23,66)(24,67)(25,130)(26,131)(27,132)(28,129)(29,48)(30,45)(31,46)(32,47)(33,71)(34,72)(35,69)(36,70)(37,53)(38,54)(39,55)(40,56)(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), (1,16,31)(2,13,32)(3,14,29)(4,15,30)(5,36,57)(6,33,58)(7,34,59)(8,35,60)(9,135,118)(10,136,119)(11,133,120)(12,134,117)(17,137,125)(18,138,126)(19,139,127)(20,140,128)(21,40,49)(22,37,50)(23,38,51)(24,39,52)(25,113,143)(26,114,144)(27,115,141)(28,116,142)(41,71,96)(42,72,93)(43,69,94)(44,70,95)(45,75,62)(46,76,63)(47,73,64)(48,74,61)(53,88,65)(54,85,66)(55,86,67)(56,87,68)(77,107,132)(78,108,129)(79,105,130)(80,106,131)(81,111,98)(82,112,99)(83,109,100)(84,110,97)(89,124,101)(90,121,102)(91,122,103)(92,123,104), (1,8,38)(2,5,39)(3,6,40)(4,7,37)(9,115,126)(10,116,127)(11,113,128)(12,114,125)(13,36,52)(14,33,49)(15,34,50)(16,35,51)(17,134,144)(18,135,141)(19,136,142)(20,133,143)(21,29,58)(22,30,59)(23,31,60)(24,32,57)(25,140,120)(26,137,117)(27,138,118)(28,139,119)(41,56,74)(42,53,75)(43,54,76)(44,55,73)(45,93,65)(46,94,66)(47,95,67)(48,96,68)(61,71,87)(62,72,88)(63,69,85)(64,70,86)(77,92,110)(78,89,111)(79,90,112)(80,91,109)(81,129,101)(82,130,102)(83,131,103)(84,132,104)(97,107,123)(98,108,124)(99,105,121)(100,106,122), (1,74)(2,75)(3,76)(4,73)(5,53)(6,54)(7,55)(8,56)(9,112)(10,109)(11,110)(12,111)(13,45)(14,46)(15,47)(16,48)(17,129)(18,130)(19,131)(20,132)(21,69)(22,70)(23,71)(24,72)(25,123)(26,124)(27,121)(28,122)(29,63)(30,64)(31,61)(32,62)(33,66)(34,67)(35,68)(36,65)(37,44)(38,41)(39,42)(40,43)(49,94)(50,95)(51,96)(52,93)(57,88)(58,85)(59,86)(60,87)(77,128)(78,125)(79,126)(80,127)(81,134)(82,135)(83,136)(84,133)(89,114)(90,115)(91,116)(92,113)(97,120)(98,117)(99,118)(100,119)(101,144)(102,141)(103,142)(104,143)(105,138)(106,139)(107,140)(108,137) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,116),(6,113),(7,114),(8,115),(13,136),(14,133),(15,134),(16,135),(17,50),(18,51),(19,52),(20,49),(21,140),(22,137),(23,138),(24,139),(25,58),(26,59),(27,60),(28,57),(29,120),(30,117),(31,118),(32,119),(33,143),(34,144),(35,141),(36,142),(37,125),(38,126),(39,127),(40,128),(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),(5,44),(6,41),(7,42),(8,43),(9,110),(10,111),(11,112),(12,109),(13,64),(14,61),(15,62),(16,63),(17,122),(18,123),(19,124),(20,121),(21,68),(22,65),(23,66),(24,67),(25,130),(26,131),(27,132),(28,129),(29,48),(30,45),(31,46),(32,47),(33,71),(34,72),(35,69),(36,70),(37,53),(38,54),(39,55),(40,56),(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)], [(1,16,31),(2,13,32),(3,14,29),(4,15,30),(5,36,57),(6,33,58),(7,34,59),(8,35,60),(9,135,118),(10,136,119),(11,133,120),(12,134,117),(17,137,125),(18,138,126),(19,139,127),(20,140,128),(21,40,49),(22,37,50),(23,38,51),(24,39,52),(25,113,143),(26,114,144),(27,115,141),(28,116,142),(41,71,96),(42,72,93),(43,69,94),(44,70,95),(45,75,62),(46,76,63),(47,73,64),(48,74,61),(53,88,65),(54,85,66),(55,86,67),(56,87,68),(77,107,132),(78,108,129),(79,105,130),(80,106,131),(81,111,98),(82,112,99),(83,109,100),(84,110,97),(89,124,101),(90,121,102),(91,122,103),(92,123,104)], [(1,8,38),(2,5,39),(3,6,40),(4,7,37),(9,115,126),(10,116,127),(11,113,128),(12,114,125),(13,36,52),(14,33,49),(15,34,50),(16,35,51),(17,134,144),(18,135,141),(19,136,142),(20,133,143),(21,29,58),(22,30,59),(23,31,60),(24,32,57),(25,140,120),(26,137,117),(27,138,118),(28,139,119),(41,56,74),(42,53,75),(43,54,76),(44,55,73),(45,93,65),(46,94,66),(47,95,67),(48,96,68),(61,71,87),(62,72,88),(63,69,85),(64,70,86),(77,92,110),(78,89,111),(79,90,112),(80,91,109),(81,129,101),(82,130,102),(83,131,103),(84,132,104),(97,107,123),(98,108,124),(99,105,121),(100,106,122)], [(1,74),(2,75),(3,76),(4,73),(5,53),(6,54),(7,55),(8,56),(9,112),(10,109),(11,110),(12,111),(13,45),(14,46),(15,47),(16,48),(17,129),(18,130),(19,131),(20,132),(21,69),(22,70),(23,71),(24,72),(25,123),(26,124),(27,121),(28,122),(29,63),(30,64),(31,61),(32,62),(33,66),(34,67),(35,68),(36,65),(37,44),(38,41),(39,42),(40,43),(49,94),(50,95),(51,96),(52,93),(57,88),(58,85),(59,86),(60,87),(77,128),(78,125),(79,126),(80,127),(81,134),(82,135),(83,136),(84,133),(89,114),(90,115),(91,116),(92,113),(97,120),(98,117),(99,118),(100,119),(101,144),(102,141),(103,142),(104,143),(105,138),(106,139),(107,140),(108,137)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4×S3 |
| kernel | C22×C4×C3⋊S3 | C2×C4×C3⋊S3 | C22×C3⋊Dic3 | C2×C6×C12 | C23×C3⋊S3 | C22×C3⋊S3 | C22×C12 | C2×C12 | C22×C6 | C2×C6 |
| # reps | 1 | 12 | 1 | 1 | 1 | 16 | 4 | 24 | 4 | 32 |
Matrix representation of C22×C4×C3⋊S3 ►in GL8(𝔽13)
| 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 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
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