direct product, non-abelian, soluble
Aliases: C32×CSU2(𝔽3), C6.20(C3×S4), (C3×C6).25S4, Q8.(S3×C32), C2.2(C32×S4), SL2(𝔽3).(C3×C6), (Q8×C32).17S3, (C3×SL2(𝔽3)).7C6, (C32×SL2(𝔽3)).3C2, (C3×Q8).10(C3×S3), SmallGroup(432,613)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C32×SL2(𝔽3) — C32×CSU2(𝔽3) |
| SL2(𝔽3) — C32×CSU2(𝔽3) |
Generators and relations for C32×CSU2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=1, d2=f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=e-1 >
Subgroups: 354 in 102 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C24, SL2(𝔽3), SL2(𝔽3), C3×Q8, C3×Q8, C33, C3×Dic3, C3×C12, C3×Q16, CSU2(𝔽3), C32×C6, C3×C24, C3×SL2(𝔽3), C3×SL2(𝔽3), Q8×C32, Q8×C32, C32×Dic3, C32×Q16, C3×CSU2(𝔽3), C32×SL2(𝔽3), C32×CSU2(𝔽3)
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, S4, CSU2(𝔽3), S3×C32, C3×S4, C3×CSU2(𝔽3), C32×S4, C32×CSU2(𝔽3)
►On 144 points
Generators in S144
(1 54 30)(2 55 31)(3 56 32)(4 53 29)(5 123 99)(6 124 100)(7 121 97)(8 122 98)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)(25 73 49)(26 74 50)(27 75 51)(28 76 52)(77 125 101)(78 126 102)(79 127 103)(80 128 104)(81 129 105)(82 130 106)(83 131 107)(84 132 108)(85 133 109)(86 134 110)(87 135 111)(88 136 112)(89 137 113)(90 138 114)(91 139 115)(92 140 116)(93 141 117)(94 142 118)(95 143 119)(96 144 120)
(1 22 14)(2 23 15)(3 24 16)(4 21 13)(5 139 131)(6 140 132)(7 137 129)(8 138 130)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 73 65)(58 74 66)(59 75 67)(60 76 68)(77 93 85)(78 94 86)(79 95 87)(80 96 88)(81 97 89)(82 98 90)(83 99 91)(84 100 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 121 113)(106 122 114)(107 123 115)(108 124 116)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(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 11 3 9)(2 10 4 12)(5 143 7 141)(6 142 8 144)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)(29 36 31 34)(30 35 32 33)(37 44 39 42)(38 43 40 41)(45 52 47 50)(46 51 48 49)(53 60 55 58)(54 59 56 57)(61 68 63 66)(62 67 64 65)(69 76 71 74)(70 75 72 73)(77 83 79 81)(78 82 80 84)(85 91 87 89)(86 90 88 92)(93 99 95 97)(94 98 96 100)(101 107 103 105)(102 106 104 108)(109 115 111 113)(110 114 112 116)(117 123 119 121)(118 122 120 124)(125 131 127 129)(126 130 128 132)(133 139 135 137)(134 138 136 140)
(2 11 10)(4 9 12)(5 8 142)(6 144 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)(29 33 36)(31 35 34)(37 41 44)(39 43 42)(45 49 52)(47 51 50)(53 57 60)(55 59 58)(61 65 68)(63 67 66)(69 73 76)(71 75 74)(78 83 82)(80 81 84)(86 91 90)(88 89 92)(94 99 98)(96 97 100)(102 107 106)(104 105 108)(110 115 114)(112 113 116)(118 123 122)(120 121 124)(126 131 130)(128 129 132)(134 139 138)(136 137 140)
(1 77 3 79)(2 81 4 83)(5 71 7 69)(6 76 8 74)(9 78 11 80)(10 84 12 82)(13 91 15 89)(14 85 16 87)(17 86 19 88)(18 92 20 90)(21 99 23 97)(22 93 24 95)(25 94 27 96)(26 100 28 98)(29 107 31 105)(30 101 32 103)(33 102 35 104)(34 108 36 106)(37 115 39 113)(38 109 40 111)(41 110 43 112)(42 116 44 114)(45 123 47 121)(46 117 48 119)(49 118 51 120)(50 124 52 122)(53 131 55 129)(54 125 56 127)(57 126 59 128)(58 132 60 130)(61 139 63 137)(62 133 64 135)(65 134 67 136)(66 140 68 138)(70 141 72 143)(73 142 75 144)
G:=sub<Sym(144)| (1,54,30)(2,55,31)(3,56,32)(4,53,29)(5,123,99)(6,124,100)(7,121,97)(8,122,98)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(25,73,49)(26,74,50)(27,75,51)(28,76,52)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,139,131)(6,140,132)(7,137,129)(8,138,130)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,11,3,9)(2,10,4,12)(5,143,7,141)(6,142,8,144)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41)(45,52,47,50)(46,51,48,49)(53,60,55,58)(54,59,56,57)(61,68,63,66)(62,67,64,65)(69,76,71,74)(70,75,72,73)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92)(93,99,95,97)(94,98,96,100)(101,107,103,105)(102,106,104,108)(109,115,111,113)(110,114,112,116)(117,123,119,121)(118,122,120,124)(125,131,127,129)(126,130,128,132)(133,139,135,137)(134,138,136,140), (2,11,10)(4,9,12)(5,8,142)(6,144,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(37,41,44)(39,43,42)(45,49,52)(47,51,50)(53,57,60)(55,59,58)(61,65,68)(63,67,66)(69,73,76)(71,75,74)(78,83,82)(80,81,84)(86,91,90)(88,89,92)(94,99,98)(96,97,100)(102,107,106)(104,105,108)(110,115,114)(112,113,116)(118,123,122)(120,121,124)(126,131,130)(128,129,132)(134,139,138)(136,137,140), (1,77,3,79)(2,81,4,83)(5,71,7,69)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,91,15,89)(14,85,16,87)(17,86,19,88)(18,92,20,90)(21,99,23,97)(22,93,24,95)(25,94,27,96)(26,100,28,98)(29,107,31,105)(30,101,32,103)(33,102,35,104)(34,108,36,106)(37,115,39,113)(38,109,40,111)(41,110,43,112)(42,116,44,114)(45,123,47,121)(46,117,48,119)(49,118,51,120)(50,124,52,122)(53,131,55,129)(54,125,56,127)(57,126,59,128)(58,132,60,130)(61,139,63,137)(62,133,64,135)(65,134,67,136)(66,140,68,138)(70,141,72,143)(73,142,75,144)>;
G:=Group( (1,54,30)(2,55,31)(3,56,32)(4,53,29)(5,123,99)(6,124,100)(7,121,97)(8,122,98)(9,57,33)(10,58,34)(11,59,35)(12,60,36)(13,61,37)(14,62,38)(15,63,39)(16,64,40)(17,65,41)(18,66,42)(19,67,43)(20,68,44)(21,69,45)(22,70,46)(23,71,47)(24,72,48)(25,73,49)(26,74,50)(27,75,51)(28,76,52)(77,125,101)(78,126,102)(79,127,103)(80,128,104)(81,129,105)(82,130,106)(83,131,107)(84,132,108)(85,133,109)(86,134,110)(87,135,111)(88,136,112)(89,137,113)(90,138,114)(91,139,115)(92,140,116)(93,141,117)(94,142,118)(95,143,119)(96,144,120), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,139,131)(6,140,132)(7,137,129)(8,138,130)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,49,41)(34,50,42)(35,51,43)(36,52,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,73,65)(58,74,66)(59,75,67)(60,76,68)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,97,89)(82,98,90)(83,99,91)(84,100,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,121,113)(106,122,114)(107,123,115)(108,124,116)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (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), (1,11,3,9)(2,10,4,12)(5,143,7,141)(6,142,8,144)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(29,36,31,34)(30,35,32,33)(37,44,39,42)(38,43,40,41)(45,52,47,50)(46,51,48,49)(53,60,55,58)(54,59,56,57)(61,68,63,66)(62,67,64,65)(69,76,71,74)(70,75,72,73)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92)(93,99,95,97)(94,98,96,100)(101,107,103,105)(102,106,104,108)(109,115,111,113)(110,114,112,116)(117,123,119,121)(118,122,120,124)(125,131,127,129)(126,130,128,132)(133,139,135,137)(134,138,136,140), (2,11,10)(4,9,12)(5,8,142)(6,144,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26)(29,33,36)(31,35,34)(37,41,44)(39,43,42)(45,49,52)(47,51,50)(53,57,60)(55,59,58)(61,65,68)(63,67,66)(69,73,76)(71,75,74)(78,83,82)(80,81,84)(86,91,90)(88,89,92)(94,99,98)(96,97,100)(102,107,106)(104,105,108)(110,115,114)(112,113,116)(118,123,122)(120,121,124)(126,131,130)(128,129,132)(134,139,138)(136,137,140), (1,77,3,79)(2,81,4,83)(5,71,7,69)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,91,15,89)(14,85,16,87)(17,86,19,88)(18,92,20,90)(21,99,23,97)(22,93,24,95)(25,94,27,96)(26,100,28,98)(29,107,31,105)(30,101,32,103)(33,102,35,104)(34,108,36,106)(37,115,39,113)(38,109,40,111)(41,110,43,112)(42,116,44,114)(45,123,47,121)(46,117,48,119)(49,118,51,120)(50,124,52,122)(53,131,55,129)(54,125,56,127)(57,126,59,128)(58,132,60,130)(61,139,63,137)(62,133,64,135)(65,134,67,136)(66,140,68,138)(70,141,72,143)(73,142,75,144) );
G=PermutationGroup([[(1,54,30),(2,55,31),(3,56,32),(4,53,29),(5,123,99),(6,124,100),(7,121,97),(8,122,98),(9,57,33),(10,58,34),(11,59,35),(12,60,36),(13,61,37),(14,62,38),(15,63,39),(16,64,40),(17,65,41),(18,66,42),(19,67,43),(20,68,44),(21,69,45),(22,70,46),(23,71,47),(24,72,48),(25,73,49),(26,74,50),(27,75,51),(28,76,52),(77,125,101),(78,126,102),(79,127,103),(80,128,104),(81,129,105),(82,130,106),(83,131,107),(84,132,108),(85,133,109),(86,134,110),(87,135,111),(88,136,112),(89,137,113),(90,138,114),(91,139,115),(92,140,116),(93,141,117),(94,142,118),(95,143,119),(96,144,120)], [(1,22,14),(2,23,15),(3,24,16),(4,21,13),(5,139,131),(6,140,132),(7,137,129),(8,138,130),(9,25,17),(10,26,18),(11,27,19),(12,28,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,49,41),(34,50,42),(35,51,43),(36,52,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,73,65),(58,74,66),(59,75,67),(60,76,68),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(81,97,89),(82,98,90),(83,99,91),(84,100,92),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,121,113),(106,122,114),(107,123,115),(108,124,116),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(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)], [(1,11,3,9),(2,10,4,12),(5,143,7,141),(6,142,8,144),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25),(29,36,31,34),(30,35,32,33),(37,44,39,42),(38,43,40,41),(45,52,47,50),(46,51,48,49),(53,60,55,58),(54,59,56,57),(61,68,63,66),(62,67,64,65),(69,76,71,74),(70,75,72,73),(77,83,79,81),(78,82,80,84),(85,91,87,89),(86,90,88,92),(93,99,95,97),(94,98,96,100),(101,107,103,105),(102,106,104,108),(109,115,111,113),(110,114,112,116),(117,123,119,121),(118,122,120,124),(125,131,127,129),(126,130,128,132),(133,139,135,137),(134,138,136,140)], [(2,11,10),(4,9,12),(5,8,142),(6,144,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26),(29,33,36),(31,35,34),(37,41,44),(39,43,42),(45,49,52),(47,51,50),(53,57,60),(55,59,58),(61,65,68),(63,67,66),(69,73,76),(71,75,74),(78,83,82),(80,81,84),(86,91,90),(88,89,92),(94,99,98),(96,97,100),(102,107,106),(104,105,108),(110,115,114),(112,113,116),(118,123,122),(120,121,124),(126,131,130),(128,129,132),(134,139,138),(136,137,140)], [(1,77,3,79),(2,81,4,83),(5,71,7,69),(6,76,8,74),(9,78,11,80),(10,84,12,82),(13,91,15,89),(14,85,16,87),(17,86,19,88),(18,92,20,90),(21,99,23,97),(22,93,24,95),(25,94,27,96),(26,100,28,98),(29,107,31,105),(30,101,32,103),(33,102,35,104),(34,108,36,106),(37,115,39,113),(38,109,40,111),(41,110,43,112),(42,116,44,114),(45,123,47,121),(46,117,48,119),(49,118,51,120),(50,124,52,122),(53,131,55,129),(54,125,56,127),(57,126,59,128),(58,132,60,130),(61,139,63,137),(62,133,64,135),(65,134,67,136),(66,140,68,138),(70,141,72,143),(73,142,75,144)]])
72 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 12 | 1 | ··· | 1 | 8 | ··· | 8 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 |
| type | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | CSU2(𝔽3) | C3×CSU2(𝔽3) | S4 | C3×S4 | CSU2(𝔽3) | C3×CSU2(𝔽3) |
| kernel | C32×CSU2(𝔽3) | C32×SL2(𝔽3) | C3×CSU2(𝔽3) | C3×SL2(𝔽3) | Q8×C32 | C3×Q8 | C32 | C3 | C3×C6 | C6 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 1 | 8 | 2 | 16 | 2 | 16 | 1 | 8 |
Matrix representation of C32×CSU2(𝔽3) ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 64 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 21 | 3 |
| 0 | 23 | 52 |
| 1 | 0 | 0 |
| 0 | 2 | 50 |
| 0 | 51 | 71 |
| 1 | 0 | 0 |
| 0 | 23 | 52 |
| 0 | 2 | 49 |
| 1 | 0 | 0 |
| 0 | 12 | 51 |
| 0 | 63 | 61 |
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,8],[64,0,0,0,1,0,0,0,1],[1,0,0,0,21,23,0,3,52],[1,0,0,0,2,51,0,50,71],[1,0,0,0,23,2,0,52,49],[1,0,0,0,12,63,0,51,61] >;
C32×CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2\times {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2xCSU(2,3)"); // GroupNames label
G:=SmallGroup(432,613);
// by ID
G=gap.SmallGroup(432,613);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,1011,3784,655,172,2273,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=e^3=1,d^2=f^2=c^2,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,d*c*d^-1=f*d*f^-1=c^-1,e*c*e^-1=c*d,f*c*f^-1=c^2*d,e*d*e^-1=c,f*e*f^-1=e^-1>;
// generators/relations