direct product, metabelian, supersoluble, monomial, rational
Aliases: C2×Q8×C3⋊S3, C62.280C23, C6⋊3(S3×Q8), (C6×Q8)⋊9S3, (C3×Q8)⋊18D6, (C2×C12).170D6, C32⋊9(C22×Q8), C6.61(S3×C23), (C3×C6).60C24, C12.112(C22×S3), (C6×C12).169C22, (C3×C12).131C23, C3⋊Dic3.49C23, (Q8×C32)⋊21C22, C32⋊4Q8⋊25C22, C3⋊4(C2×S3×Q8), (Q8×C3×C6)⋊12C2, (C3×C6)⋊8(C2×Q8), C2.9(C23×C3⋊S3), C4.22(C22×C3⋊S3), (C4×C3⋊S3).77C22, (C2×C3⋊S3).60C23, (C2×C32⋊4Q8)⋊22C2, (C2×C6).288(C22×S3), C22.31(C22×C3⋊S3), (C22×C3⋊S3).114C22, (C2×C3⋊Dic3).183C22, (C2×C4×C3⋊S3).10C2, (C2×C4).61(C2×C3⋊S3), SmallGroup(288,1010)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×C3⋊S3
G = < a,b,c,d,e,f | a2=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: 1412 in 468 conjugacy classes, 173 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, Q8, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C2×Q8, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C22×S3, C22×Q8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, Q8×C32, C22×C3⋊S3, C2×S3×Q8, C2×C32⋊4Q8, C2×C4×C3⋊S3, Q8×C3⋊S3, Q8×C3×C6, C2×Q8×C3⋊S3
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, C3⋊S3, C22×S3, C22×Q8, C2×C3⋊S3, S3×Q8, S3×C23, C22×C3⋊S3, C2×S3×Q8, Q8×C3⋊S3, C23×C3⋊S3, C2×Q8×C3⋊S3
►On 144 points
Generators in S144
(1 112)(2 109)(3 110)(4 111)(5 87)(6 88)(7 85)(8 86)(9 80)(10 77)(11 78)(12 79)(13 98)(14 99)(15 100)(16 97)(17 75)(18 76)(19 73)(20 74)(21 83)(22 84)(23 81)(24 82)(25 96)(26 93)(27 94)(28 95)(29 91)(30 92)(31 89)(32 90)(33 105)(34 106)(35 107)(36 108)(37 101)(38 102)(39 103)(40 104)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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 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)(9 115 141)(10 116 142)(11 113 143)(12 114 144)(13 59 37)(14 60 38)(15 57 39)(16 58 40)(17 125 117)(18 126 118)(19 127 119)(20 128 120)(25 140 133)(26 137 134)(27 138 135)(28 139 136)(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 89 110)(82 90 111)(83 91 112)(84 92 109)(85 108 124)(86 105 121)(87 106 122)(88 107 123)(97 130 104)(98 131 101)(99 132 102)(100 129 103)
(1 8 16)(2 5 13)(3 6 14)(4 7 15)(9 138 118)(10 139 119)(11 140 120)(12 137 117)(17 114 134)(18 115 135)(19 116 136)(20 113 133)(21 33 58)(22 34 59)(23 35 60)(24 36 57)(25 128 143)(26 125 144)(27 126 141)(28 127 142)(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)(81 107 132)(82 108 129)(83 105 130)(84 106 131)(85 100 111)(86 97 112)(87 98 109)(88 99 110)(89 123 102)(90 124 103)(91 121 104)(92 122 101)
(1 110)(2 111)(3 112)(4 109)(5 100)(6 97)(7 98)(8 99)(9 96)(10 93)(11 94)(12 95)(13 85)(14 86)(15 87)(16 88)(17 73)(18 74)(19 75)(20 76)(21 89)(22 90)(23 91)(24 92)(25 80)(26 77)(27 78)(28 79)(29 81)(30 82)(31 83)(32 84)(33 102)(34 103)(35 104)(36 101)(37 108)(38 105)(39 106)(40 107)(41 135)(42 136)(43 133)(44 134)(45 127)(46 128)(47 125)(48 126)(49 132)(50 129)(51 130)(52 131)(53 119)(54 120)(55 117)(56 118)(57 122)(58 123)(59 124)(60 121)(61 115)(62 116)(63 113)(64 114)(65 142)(66 143)(67 144)(68 141)(69 140)(70 137)(71 138)(72 139)
G:=sub<Sym(144)| (1,112)(2,109)(3,110)(4,111)(5,87)(6,88)(7,85)(8,86)(9,80)(10,77)(11,78)(12,79)(13,98)(14,99)(15,100)(16,97)(17,75)(18,76)(19,73)(20,74)(21,83)(22,84)(23,81)(24,82)(25,96)(26,93)(27,94)(28,95)(29,91)(30,92)(31,89)(32,90)(33,105)(34,106)(35,107)(36,108)(37,101)(38,102)(39,103)(40,104)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,76,3,74)(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)(9,115,141)(10,116,142)(11,113,143)(12,114,144)(13,59,37)(14,60,38)(15,57,39)(16,58,40)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(25,140,133)(26,137,134)(27,138,135)(28,139,136)(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)(81,89,110)(82,90,111)(83,91,112)(84,92,109)(85,108,124)(86,105,121)(87,106,122)(88,107,123)(97,130,104)(98,131,101)(99,132,102)(100,129,103), (1,8,16)(2,5,13)(3,6,14)(4,7,15)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,33,58)(22,34,59)(23,35,60)(24,36,57)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(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)(81,107,132)(82,108,129)(83,105,130)(84,106,131)(85,100,111)(86,97,112)(87,98,109)(88,99,110)(89,123,102)(90,124,103)(91,121,104)(92,122,101), (1,110)(2,111)(3,112)(4,109)(5,100)(6,97)(7,98)(8,99)(9,96)(10,93)(11,94)(12,95)(13,85)(14,86)(15,87)(16,88)(17,73)(18,74)(19,75)(20,76)(21,89)(22,90)(23,91)(24,92)(25,80)(26,77)(27,78)(28,79)(29,81)(30,82)(31,83)(32,84)(33,102)(34,103)(35,104)(36,101)(37,108)(38,105)(39,106)(40,107)(41,135)(42,136)(43,133)(44,134)(45,127)(46,128)(47,125)(48,126)(49,132)(50,129)(51,130)(52,131)(53,119)(54,120)(55,117)(56,118)(57,122)(58,123)(59,124)(60,121)(61,115)(62,116)(63,113)(64,114)(65,142)(66,143)(67,144)(68,141)(69,140)(70,137)(71,138)(72,139)>;
G:=Group( (1,112)(2,109)(3,110)(4,111)(5,87)(6,88)(7,85)(8,86)(9,80)(10,77)(11,78)(12,79)(13,98)(14,99)(15,100)(16,97)(17,75)(18,76)(19,73)(20,74)(21,83)(22,84)(23,81)(24,82)(25,96)(26,93)(27,94)(28,95)(29,91)(30,92)(31,89)(32,90)(33,105)(34,106)(35,107)(36,108)(37,101)(38,102)(39,103)(40,104)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(55,127)(56,128)(57,129)(58,130)(59,131)(60,132)(61,133)(62,134)(63,135)(64,136)(65,137)(66,138)(67,139)(68,140)(69,141)(70,142)(71,143)(72,144), (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), (1,76,3,74)(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)(9,115,141)(10,116,142)(11,113,143)(12,114,144)(13,59,37)(14,60,38)(15,57,39)(16,58,40)(17,125,117)(18,126,118)(19,127,119)(20,128,120)(25,140,133)(26,137,134)(27,138,135)(28,139,136)(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)(81,89,110)(82,90,111)(83,91,112)(84,92,109)(85,108,124)(86,105,121)(87,106,122)(88,107,123)(97,130,104)(98,131,101)(99,132,102)(100,129,103), (1,8,16)(2,5,13)(3,6,14)(4,7,15)(9,138,118)(10,139,119)(11,140,120)(12,137,117)(17,114,134)(18,115,135)(19,116,136)(20,113,133)(21,33,58)(22,34,59)(23,35,60)(24,36,57)(25,128,143)(26,125,144)(27,126,141)(28,127,142)(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)(81,107,132)(82,108,129)(83,105,130)(84,106,131)(85,100,111)(86,97,112)(87,98,109)(88,99,110)(89,123,102)(90,124,103)(91,121,104)(92,122,101), (1,110)(2,111)(3,112)(4,109)(5,100)(6,97)(7,98)(8,99)(9,96)(10,93)(11,94)(12,95)(13,85)(14,86)(15,87)(16,88)(17,73)(18,74)(19,75)(20,76)(21,89)(22,90)(23,91)(24,92)(25,80)(26,77)(27,78)(28,79)(29,81)(30,82)(31,83)(32,84)(33,102)(34,103)(35,104)(36,101)(37,108)(38,105)(39,106)(40,107)(41,135)(42,136)(43,133)(44,134)(45,127)(46,128)(47,125)(48,126)(49,132)(50,129)(51,130)(52,131)(53,119)(54,120)(55,117)(56,118)(57,122)(58,123)(59,124)(60,121)(61,115)(62,116)(63,113)(64,114)(65,142)(66,143)(67,144)(68,141)(69,140)(70,137)(71,138)(72,139) );
G=PermutationGroup([[(1,112),(2,109),(3,110),(4,111),(5,87),(6,88),(7,85),(8,86),(9,80),(10,77),(11,78),(12,79),(13,98),(14,99),(15,100),(16,97),(17,75),(18,76),(19,73),(20,74),(21,83),(22,84),(23,81),(24,82),(25,96),(26,93),(27,94),(28,95),(29,91),(30,92),(31,89),(32,90),(33,105),(34,106),(35,107),(36,108),(37,101),(38,102),(39,103),(40,104),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(55,127),(56,128),(57,129),(58,130),(59,131),(60,132),(61,133),(62,134),(63,135),(64,136),(65,137),(66,138),(67,139),(68,140),(69,141),(70,142),(71,143),(72,144)], [(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)], [(1,76,3,74),(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),(9,115,141),(10,116,142),(11,113,143),(12,114,144),(13,59,37),(14,60,38),(15,57,39),(16,58,40),(17,125,117),(18,126,118),(19,127,119),(20,128,120),(25,140,133),(26,137,134),(27,138,135),(28,139,136),(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),(81,89,110),(82,90,111),(83,91,112),(84,92,109),(85,108,124),(86,105,121),(87,106,122),(88,107,123),(97,130,104),(98,131,101),(99,132,102),(100,129,103)], [(1,8,16),(2,5,13),(3,6,14),(4,7,15),(9,138,118),(10,139,119),(11,140,120),(12,137,117),(17,114,134),(18,115,135),(19,116,136),(20,113,133),(21,33,58),(22,34,59),(23,35,60),(24,36,57),(25,128,143),(26,125,144),(27,126,141),(28,127,142),(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),(81,107,132),(82,108,129),(83,105,130),(84,106,131),(85,100,111),(86,97,112),(87,98,109),(88,99,110),(89,123,102),(90,124,103),(91,121,104),(92,122,101)], [(1,110),(2,111),(3,112),(4,109),(5,100),(6,97),(7,98),(8,99),(9,96),(10,93),(11,94),(12,95),(13,85),(14,86),(15,87),(16,88),(17,73),(18,74),(19,75),(20,76),(21,89),(22,90),(23,91),(24,92),(25,80),(26,77),(27,78),(28,79),(29,81),(30,82),(31,83),(32,84),(33,102),(34,103),(35,104),(36,101),(37,108),(38,105),(39,106),(40,107),(41,135),(42,136),(43,133),(44,134),(45,127),(46,128),(47,125),(48,126),(49,132),(50,129),(51,130),(52,131),(53,119),(54,120),(55,117),(56,118),(57,122),(58,123),(59,124),(60,121),(61,115),(62,116),(63,113),(64,114),(65,142),(66,143),(67,144),(68,141),(69,140),(70,137),(71,138),(72,139)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4F | 4G | ··· | 4L | 6A | ··· | 6L | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | S3×Q8 |
| kernel | C2×Q8×C3⋊S3 | C2×C32⋊4Q8 | C2×C4×C3⋊S3 | Q8×C3⋊S3 | Q8×C3×C6 | C6×Q8 | C2×C3⋊S3 | C2×C12 | C3×Q8 | C6 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 4 | 12 | 16 | 8 |
Matrix representation of C2×Q8×C3⋊S3 ►in GL6(𝔽13)
| 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 0 | 8 | 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 | 7 | 4 |
| 0 | 0 | 0 | 0 | 7 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,8,0,0,0,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,7,7,0,0,0,0,4,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×Q8×C3⋊S3 in GAP, Magma, Sage, TeX
C_2\times Q_8\times C_3\rtimes S_3
% in TeX
G:=Group("C2xQ8xC3:S3"); // GroupNames label
G:=SmallGroup(288,1010);
// by ID
G=gap.SmallGroup(288,1010);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,185,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=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