direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8×C3⋊S3, C33⋊20(C2×Q8), C12.40(S3×C6), (Q8×C33)⋊6C2, C32⋊15(S3×Q8), C32⋊11(C6×Q8), (C3×C12).150D6, (Q8×C32)⋊14S3, (Q8×C32)⋊16C6, C32⋊4Q8⋊13C6, (C32×C6).91C23, (C32×C12).55C22, C3⋊3(C3×S3×Q8), C4.6(C6×C3⋊S3), C6.58(S3×C2×C6), (C4×C3⋊S3).5C6, (C3×Q8)⋊6(C3×S3), (C12×C3⋊S3).8C2, C12.57(C2×C3⋊S3), (C3×C12).59(C2×C6), C6.58(C22×C3⋊S3), (C6×C3⋊S3).67C22, C3⋊Dic3.24(C2×C6), (C3×C6).65(C22×C6), (C3×C32⋊4Q8)⋊15C2, (C3×C6).180(C22×S3), (C3×C3⋊Dic3).61C22, C2.8(C2×C6×C3⋊S3), (C2×C3⋊S3).27(C2×C6), SmallGroup(432,716)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8×C3⋊S3
G = < a,b,c,d,e,f | a3=b4=d3=e3=f2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc-1=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=d-1, fef=e-1 >
Subgroups: 740 in 276 conjugacy classes, 98 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, Q8, Q8, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×Q8, C6×Q8, C3×C3⋊S3, C32×C6, C3×Dic6, S3×C12, C32⋊4Q8, C4×C3⋊S3, Q8×C32, Q8×C32, Q8×C32, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×S3×Q8, Q8×C3⋊S3, C3×C32⋊4Q8, C12×C3⋊S3, Q8×C33, C3×Q8×C3⋊S3
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×Q8, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, S3×Q8, C6×Q8, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×S3×Q8, Q8×C3⋊S3, C2×C6×C3⋊S3, C3×Q8×C3⋊S3
►On 144 points
Generators in S144
(1 33 40)(2 34 37)(3 35 38)(4 36 39)(5 59 30)(6 60 31)(7 57 32)(8 58 29)(9 135 126)(10 136 127)(11 133 128)(12 134 125)(13 22 50)(14 23 51)(15 24 52)(16 21 49)(17 144 137)(18 141 138)(19 142 139)(20 143 140)(25 120 113)(26 117 114)(27 118 115)(28 119 116)(41 96 48)(42 93 45)(43 94 46)(44 95 47)(53 79 62)(54 80 63)(55 77 64)(56 78 61)(65 75 72)(66 76 69)(67 73 70)(68 74 71)(81 123 99)(82 124 100)(83 121 97)(84 122 98)(85 129 90)(86 130 91)(87 131 92)(88 132 89)(101 109 106)(102 110 107)(103 111 108)(104 112 105)
(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 76 3 74)(2 75 4 73)(5 42 7 44)(6 41 8 43)(9 123 11 121)(10 122 12 124)(13 62 15 64)(14 61 16 63)(17 111 19 109)(18 110 20 112)(21 54 23 56)(22 53 24 55)(25 130 27 132)(26 129 28 131)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 65 39 67)(38 68 40 66)(49 80 51 78)(50 79 52 77)(57 95 59 93)(58 94 60 96)(81 128 83 126)(82 127 84 125)(85 116 87 114)(86 115 88 113)(89 120 91 118)(90 119 92 117)(97 135 99 133)(98 134 100 136)(101 137 103 139)(102 140 104 138)(105 141 107 143)(106 144 108 142)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 34 50)(6 35 51)(7 36 52)(8 33 49)(9 141 115)(10 142 116)(11 143 113)(12 144 114)(13 59 37)(14 60 38)(15 57 39)(16 58 40)(17 117 125)(18 118 126)(19 119 127)(20 120 128)(25 133 140)(26 134 137)(27 135 138)(28 136 139)(41 71 78)(42 72 79)(43 69 80)(44 70 77)(45 75 53)(46 76 54)(47 73 55)(48 74 56)(61 96 68)(62 93 65)(63 94 66)(64 95 67)(81 110 89)(82 111 90)(83 112 91)(84 109 92)(85 124 108)(86 121 105)(87 122 106)(88 123 107)(97 104 130)(98 101 131)(99 102 132)(100 103 129)
(1 8 16)(2 5 13)(3 6 14)(4 7 15)(9 118 138)(10 119 139)(11 120 140)(12 117 137)(17 134 114)(18 135 115)(19 136 116)(20 133 113)(21 33 58)(22 34 59)(23 35 60)(24 36 57)(25 143 128)(26 144 125)(27 141 126)(28 142 127)(29 49 40)(30 50 37)(31 51 38)(32 52 39)(41 61 74)(42 62 75)(43 63 76)(44 64 73)(45 79 65)(46 80 66)(47 77 67)(48 78 68)(53 72 93)(54 69 94)(55 70 95)(56 71 96)(81 132 107)(82 129 108)(83 130 105)(84 131 106)(85 111 100)(86 112 97)(87 109 98)(88 110 99)(89 102 123)(90 103 124)(91 104 121)(92 101 122)
(1 110)(2 111)(3 112)(4 109)(5 85)(6 86)(7 87)(8 88)(9 78)(10 79)(11 80)(12 77)(13 100)(14 97)(15 98)(16 99)(17 73)(18 74)(19 75)(20 76)(21 81)(22 82)(23 83)(24 84)(25 94)(26 95)(27 96)(28 93)(29 89)(30 90)(31 91)(32 92)(33 107)(34 108)(35 105)(36 106)(37 103)(38 104)(39 101)(40 102)(41 115)(42 116)(43 113)(44 114)(45 119)(46 120)(47 117)(48 118)(49 123)(50 124)(51 121)(52 122)(53 127)(54 128)(55 125)(56 126)(57 131)(58 132)(59 129)(60 130)(61 135)(62 136)(63 133)(64 134)(65 139)(66 140)(67 137)(68 138)(69 143)(70 144)(71 141)(72 142)
G:=sub<Sym(144)| (1,33,40)(2,34,37)(3,35,38)(4,36,39)(5,59,30)(6,60,31)(7,57,32)(8,58,29)(9,135,126)(10,136,127)(11,133,128)(12,134,125)(13,22,50)(14,23,51)(15,24,52)(16,21,49)(17,144,137)(18,141,138)(19,142,139)(20,143,140)(25,120,113)(26,117,114)(27,118,115)(28,119,116)(41,96,48)(42,93,45)(43,94,46)(44,95,47)(53,79,62)(54,80,63)(55,77,64)(56,78,61)(65,75,72)(66,76,69)(67,73,70)(68,74,71)(81,123,99)(82,124,100)(83,121,97)(84,122,98)(85,129,90)(86,130,91)(87,131,92)(88,132,89)(101,109,106)(102,110,107)(103,111,108)(104,112,105), (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), (1,76,3,74)(2,75,4,73)(5,42,7,44)(6,41,8,43)(9,123,11,121)(10,122,12,124)(13,62,15,64)(14,61,16,63)(17,111,19,109)(18,110,20,112)(21,54,23,56)(22,53,24,55)(25,130,27,132)(26,129,28,131)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,65,39,67)(38,68,40,66)(49,80,51,78)(50,79,52,77)(57,95,59,93)(58,94,60,96)(81,128,83,126)(82,127,84,125)(85,116,87,114)(86,115,88,113)(89,120,91,118)(90,119,92,117)(97,135,99,133)(98,134,100,136)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,34,50)(6,35,51)(7,36,52)(8,33,49)(9,141,115)(10,142,116)(11,143,113)(12,144,114)(13,59,37)(14,60,38)(15,57,39)(16,58,40)(17,117,125)(18,118,126)(19,119,127)(20,120,128)(25,133,140)(26,134,137)(27,135,138)(28,136,139)(41,71,78)(42,72,79)(43,69,80)(44,70,77)(45,75,53)(46,76,54)(47,73,55)(48,74,56)(61,96,68)(62,93,65)(63,94,66)(64,95,67)(81,110,89)(82,111,90)(83,112,91)(84,109,92)(85,124,108)(86,121,105)(87,122,106)(88,123,107)(97,104,130)(98,101,131)(99,102,132)(100,103,129), (1,8,16)(2,5,13)(3,6,14)(4,7,15)(9,118,138)(10,119,139)(11,120,140)(12,117,137)(17,134,114)(18,135,115)(19,136,116)(20,133,113)(21,33,58)(22,34,59)(23,35,60)(24,36,57)(25,143,128)(26,144,125)(27,141,126)(28,142,127)(29,49,40)(30,50,37)(31,51,38)(32,52,39)(41,61,74)(42,62,75)(43,63,76)(44,64,73)(45,79,65)(46,80,66)(47,77,67)(48,78,68)(53,72,93)(54,69,94)(55,70,95)(56,71,96)(81,132,107)(82,129,108)(83,130,105)(84,131,106)(85,111,100)(86,112,97)(87,109,98)(88,110,99)(89,102,123)(90,103,124)(91,104,121)(92,101,122), (1,110)(2,111)(3,112)(4,109)(5,85)(6,86)(7,87)(8,88)(9,78)(10,79)(11,80)(12,77)(13,100)(14,97)(15,98)(16,99)(17,73)(18,74)(19,75)(20,76)(21,81)(22,82)(23,83)(24,84)(25,94)(26,95)(27,96)(28,93)(29,89)(30,90)(31,91)(32,92)(33,107)(34,108)(35,105)(36,106)(37,103)(38,104)(39,101)(40,102)(41,115)(42,116)(43,113)(44,114)(45,119)(46,120)(47,117)(48,118)(49,123)(50,124)(51,121)(52,122)(53,127)(54,128)(55,125)(56,126)(57,131)(58,132)(59,129)(60,130)(61,135)(62,136)(63,133)(64,134)(65,139)(66,140)(67,137)(68,138)(69,143)(70,144)(71,141)(72,142)>;
G:=Group( (1,33,40)(2,34,37)(3,35,38)(4,36,39)(5,59,30)(6,60,31)(7,57,32)(8,58,29)(9,135,126)(10,136,127)(11,133,128)(12,134,125)(13,22,50)(14,23,51)(15,24,52)(16,21,49)(17,144,137)(18,141,138)(19,142,139)(20,143,140)(25,120,113)(26,117,114)(27,118,115)(28,119,116)(41,96,48)(42,93,45)(43,94,46)(44,95,47)(53,79,62)(54,80,63)(55,77,64)(56,78,61)(65,75,72)(66,76,69)(67,73,70)(68,74,71)(81,123,99)(82,124,100)(83,121,97)(84,122,98)(85,129,90)(86,130,91)(87,131,92)(88,132,89)(101,109,106)(102,110,107)(103,111,108)(104,112,105), (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), (1,76,3,74)(2,75,4,73)(5,42,7,44)(6,41,8,43)(9,123,11,121)(10,122,12,124)(13,62,15,64)(14,61,16,63)(17,111,19,109)(18,110,20,112)(21,54,23,56)(22,53,24,55)(25,130,27,132)(26,129,28,131)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,65,39,67)(38,68,40,66)(49,80,51,78)(50,79,52,77)(57,95,59,93)(58,94,60,96)(81,128,83,126)(82,127,84,125)(85,116,87,114)(86,115,88,113)(89,120,91,118)(90,119,92,117)(97,135,99,133)(98,134,100,136)(101,137,103,139)(102,140,104,138)(105,141,107,143)(106,144,108,142), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,34,50)(6,35,51)(7,36,52)(8,33,49)(9,141,115)(10,142,116)(11,143,113)(12,144,114)(13,59,37)(14,60,38)(15,57,39)(16,58,40)(17,117,125)(18,118,126)(19,119,127)(20,120,128)(25,133,140)(26,134,137)(27,135,138)(28,136,139)(41,71,78)(42,72,79)(43,69,80)(44,70,77)(45,75,53)(46,76,54)(47,73,55)(48,74,56)(61,96,68)(62,93,65)(63,94,66)(64,95,67)(81,110,89)(82,111,90)(83,112,91)(84,109,92)(85,124,108)(86,121,105)(87,122,106)(88,123,107)(97,104,130)(98,101,131)(99,102,132)(100,103,129), (1,8,16)(2,5,13)(3,6,14)(4,7,15)(9,118,138)(10,119,139)(11,120,140)(12,117,137)(17,134,114)(18,135,115)(19,136,116)(20,133,113)(21,33,58)(22,34,59)(23,35,60)(24,36,57)(25,143,128)(26,144,125)(27,141,126)(28,142,127)(29,49,40)(30,50,37)(31,51,38)(32,52,39)(41,61,74)(42,62,75)(43,63,76)(44,64,73)(45,79,65)(46,80,66)(47,77,67)(48,78,68)(53,72,93)(54,69,94)(55,70,95)(56,71,96)(81,132,107)(82,129,108)(83,130,105)(84,131,106)(85,111,100)(86,112,97)(87,109,98)(88,110,99)(89,102,123)(90,103,124)(91,104,121)(92,101,122), (1,110)(2,111)(3,112)(4,109)(5,85)(6,86)(7,87)(8,88)(9,78)(10,79)(11,80)(12,77)(13,100)(14,97)(15,98)(16,99)(17,73)(18,74)(19,75)(20,76)(21,81)(22,82)(23,83)(24,84)(25,94)(26,95)(27,96)(28,93)(29,89)(30,90)(31,91)(32,92)(33,107)(34,108)(35,105)(36,106)(37,103)(38,104)(39,101)(40,102)(41,115)(42,116)(43,113)(44,114)(45,119)(46,120)(47,117)(48,118)(49,123)(50,124)(51,121)(52,122)(53,127)(54,128)(55,125)(56,126)(57,131)(58,132)(59,129)(60,130)(61,135)(62,136)(63,133)(64,134)(65,139)(66,140)(67,137)(68,138)(69,143)(70,144)(71,141)(72,142) );
G=PermutationGroup([[(1,33,40),(2,34,37),(3,35,38),(4,36,39),(5,59,30),(6,60,31),(7,57,32),(8,58,29),(9,135,126),(10,136,127),(11,133,128),(12,134,125),(13,22,50),(14,23,51),(15,24,52),(16,21,49),(17,144,137),(18,141,138),(19,142,139),(20,143,140),(25,120,113),(26,117,114),(27,118,115),(28,119,116),(41,96,48),(42,93,45),(43,94,46),(44,95,47),(53,79,62),(54,80,63),(55,77,64),(56,78,61),(65,75,72),(66,76,69),(67,73,70),(68,74,71),(81,123,99),(82,124,100),(83,121,97),(84,122,98),(85,129,90),(86,130,91),(87,131,92),(88,132,89),(101,109,106),(102,110,107),(103,111,108),(104,112,105)], [(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)], [(1,76,3,74),(2,75,4,73),(5,42,7,44),(6,41,8,43),(9,123,11,121),(10,122,12,124),(13,62,15,64),(14,61,16,63),(17,111,19,109),(18,110,20,112),(21,54,23,56),(22,53,24,55),(25,130,27,132),(26,129,28,131),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,65,39,67),(38,68,40,66),(49,80,51,78),(50,79,52,77),(57,95,59,93),(58,94,60,96),(81,128,83,126),(82,127,84,125),(85,116,87,114),(86,115,88,113),(89,120,91,118),(90,119,92,117),(97,135,99,133),(98,134,100,136),(101,137,103,139),(102,140,104,138),(105,141,107,143),(106,144,108,142)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,34,50),(6,35,51),(7,36,52),(8,33,49),(9,141,115),(10,142,116),(11,143,113),(12,144,114),(13,59,37),(14,60,38),(15,57,39),(16,58,40),(17,117,125),(18,118,126),(19,119,127),(20,120,128),(25,133,140),(26,134,137),(27,135,138),(28,136,139),(41,71,78),(42,72,79),(43,69,80),(44,70,77),(45,75,53),(46,76,54),(47,73,55),(48,74,56),(61,96,68),(62,93,65),(63,94,66),(64,95,67),(81,110,89),(82,111,90),(83,112,91),(84,109,92),(85,124,108),(86,121,105),(87,122,106),(88,123,107),(97,104,130),(98,101,131),(99,102,132),(100,103,129)], [(1,8,16),(2,5,13),(3,6,14),(4,7,15),(9,118,138),(10,119,139),(11,120,140),(12,117,137),(17,134,114),(18,135,115),(19,136,116),(20,133,113),(21,33,58),(22,34,59),(23,35,60),(24,36,57),(25,143,128),(26,144,125),(27,141,126),(28,142,127),(29,49,40),(30,50,37),(31,51,38),(32,52,39),(41,61,74),(42,62,75),(43,63,76),(44,64,73),(45,79,65),(46,80,66),(47,77,67),(48,78,68),(53,72,93),(54,69,94),(55,70,95),(56,71,96),(81,132,107),(82,129,108),(83,130,105),(84,131,106),(85,111,100),(86,112,97),(87,109,98),(88,110,99),(89,102,123),(90,103,124),(91,104,121),(92,101,122)], [(1,110),(2,111),(3,112),(4,109),(5,85),(6,86),(7,87),(8,88),(9,78),(10,79),(11,80),(12,77),(13,100),(14,97),(15,98),(16,99),(17,73),(18,74),(19,75),(20,76),(21,81),(22,82),(23,83),(24,84),(25,94),(26,95),(27,96),(28,93),(29,89),(30,90),(31,91),(32,92),(33,107),(34,108),(35,105),(36,106),(37,103),(38,104),(39,101),(40,102),(41,115),(42,116),(43,113),(44,114),(45,119),(46,120),(47,117),(48,118),(49,123),(50,124),(51,121),(52,122),(53,127),(54,128),(55,125),(56,126),(57,131),(58,132),(59,129),(60,130),(61,135),(62,136),(63,133),(64,134),(65,139),(66,140),(67,137),(68,138),(69,143),(70,144),(71,141),(72,142)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 6Q | 6R | 12A | ··· | 12F | 12G | ··· | 12AP | 12AQ | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 2 | ··· | 2 | 4 | ··· | 4 | 18 | ··· | 18 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | C3×S3 | C3×Q8 | S3×C6 | S3×Q8 | C3×S3×Q8 |
| kernel | C3×Q8×C3⋊S3 | C3×C32⋊4Q8 | C12×C3⋊S3 | Q8×C33 | Q8×C3⋊S3 | C32⋊4Q8 | C4×C3⋊S3 | Q8×C32 | Q8×C32 | C3×C3⋊S3 | C3×C12 | C3×Q8 | C3⋊S3 | C12 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 4 | 2 | 12 | 8 | 4 | 24 | 4 | 8 |
Matrix representation of C3×Q8×C3⋊S3 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 10 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 3 | 9 | 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 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,10,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,1,10],[9,10,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C3×Q8×C3⋊S3 in GAP, Magma, Sage, TeX
C_3\times Q_8\times C_3\rtimes S_3
% in TeX
G:=Group("C3xQ8xC3:S3"); // GroupNames label
G:=SmallGroup(432,716);
// by ID
G=gap.SmallGroup(432,716);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^4=d^3=e^3=f^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=b^-1,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