G = C32.CSU2(𝔽3) order 432 = 24·33
The non-split extension by C32 of CSU2(𝔽3) acting via CSU2(𝔽3)/Q8=S3
Aliases: C32.CSU2(𝔽3), Q8⋊C9.C6, Q8.D9⋊C3, C6.1(C3×S4), (C3×C6).1S4, Q8.(C9⋊C6), (Q8×C32).5S3, C2.2(C32.S4), Q8⋊3- 1+2.C2, C3.1(C3×CSU2(𝔽3)), (C3×Q8).1(C3×S3), SmallGroup(432,243)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C32.CSU2(𝔽3) |
Generators and relations for C32.CSU2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=1, d2=f2=c2, e3=fbf-1=b-1, eae-1=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, be=eb, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=be2 >
Character table of C32.CSU2(𝔽3)
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 18A | 18B | 18C | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 3 | 3 | 6 | 36 | 2 | 3 | 3 | 18 | 18 | 24 | 24 | 24 | 6 | 6 | 12 | 12 | 12 | 36 | 36 | 24 | 24 | 24 | 18 | 18 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ3 | 1 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ32 | 1 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | √2 | -√2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | √2 | -√2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -√2 | √2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -√2 | √2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | ζ6 | ζ65 | -1 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ11 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | ζ65 | ζ6 | -1 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ12 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | √2 | -√2 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 1 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | complex lifted from C3×CSU2(𝔽3) |
| ρ13 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | -√2 | √2 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 1 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | complex lifted from C3×CSU2(𝔽3) |
| ρ14 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | -√2 | √2 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | 1 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | complex lifted from C3×CSU2(𝔽3) |
| ρ15 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | √2 | -√2 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | 1 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | complex lifted from C3×CSU2(𝔽3) |
| ρ16 | 3 | 3 | 3 | 3 | 3 | -1 | -1 | 3 | 3 | 3 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 3 | 3 | 3 | -1 | 1 | 3 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 1 | 1 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | 0 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from C3×S4 |
| ρ19 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | ζ3 | ζ32 | 0 | 0 | 0 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from C3×S4 |
| ρ20 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 1 | 1 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | 0 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from C3×S4 |
| ρ21 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | ζ32 | ζ3 | 0 | 0 | 0 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from C3×S4 |
| ρ22 | 4 | -4 | 4 | 4 | 4 | 0 | 0 | -4 | -4 | -4 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ23 | 4 | -4 | 4 | -2+2√-3 | -2-2√-3 | 0 | 0 | -4 | 2+2√-3 | 2-2√-3 | 0 | 0 | ζ3 | ζ32 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×CSU2(𝔽3) |
| ρ24 | 4 | -4 | 4 | -2-2√-3 | -2+2√-3 | 0 | 0 | -4 | 2-2√-3 | 2+2√-3 | 0 | 0 | ζ32 | ζ3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -1 | 0 | 0 | 0 | 0 | complex lifted from C3×CSU2(𝔽3) |
| ρ25 | 6 | 6 | -3 | 0 | 0 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ26 | 6 | 6 | -3 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32.S4 |
| ρ27 | 6 | 6 | -3 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2+2√-3 | -2-2√-3 | 1 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32.S4 |
| ρ28 | 6 | 6 | -3 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2-2√-3 | -2+2√-3 | 1 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32.S4 |
| ρ29 | 12 | -12 | -6 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 144 points
Generators in S144
(2 8 5)(3 6 9)(10 13 16)(12 18 15)(19 25 22)(20 23 26)(29 35 32)(30 33 36)(38 44 41)(39 42 45)(46 49 52)(48 54 51)(56 62 59)(57 60 63)(65 71 68)(66 69 72)(73 79 76)(74 77 80)(82 88 85)(83 86 89)(91 94 97)(93 99 96)(100 106 103)(101 104 107)(109 115 112)(110 113 116)(119 125 122)(120 123 126)(127 130 133)(129 135 132)(136 142 139)(137 140 143)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)
(1 87 131 102)(2 109 132 99)(3 77 133 123)(4 90 134 105)(5 112 135 93)(6 80 127 126)(7 84 128 108)(8 115 129 96)(9 74 130 120)(10 63 143 39)(11 67 144 21)(12 35 136 54)(13 57 137 42)(14 70 138 24)(15 29 139 48)(16 60 140 45)(17 64 141 27)(18 32 142 51)(19 62 65 38)(20 52 66 33)(22 56 68 41)(23 46 69 36)(25 59 71 44)(26 49 72 30)(28 58 47 43)(31 61 50 37)(34 55 53 40)(73 85 119 100)(75 98 121 117)(76 88 122 103)(78 92 124 111)(79 82 125 106)(81 95 118 114)(83 113 107 94)(86 116 101 97)(89 110 104 91)
(1 75 131 121)(2 88 132 103)(3 110 133 91)(4 78 134 124)(5 82 135 106)(6 113 127 94)(7 81 128 118)(8 85 129 100)(9 116 130 97)(10 33 143 52)(11 55 144 40)(12 68 136 22)(13 36 137 46)(14 58 138 43)(15 71 139 25)(16 30 140 49)(17 61 141 37)(18 65 142 19)(20 63 66 39)(21 53 67 34)(23 57 69 42)(24 47 70 28)(26 60 72 45)(27 50 64 31)(29 59 48 44)(32 62 51 38)(35 56 54 41)(73 96 119 115)(74 86 120 101)(76 99 122 109)(77 89 123 104)(79 93 125 112)(80 83 126 107)(84 114 108 95)(87 117 102 98)(90 111 105 92)
(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 14 131 138)(2 13 132 137)(3 12 133 136)(4 11 134 144)(5 10 135 143)(6 18 127 142)(7 17 128 141)(8 16 129 140)(9 15 130 139)(19 126 65 80)(20 125 66 79)(21 124 67 78)(22 123 68 77)(23 122 69 76)(24 121 70 75)(25 120 71 74)(26 119 72 73)(27 118 64 81)(28 117 47 98)(29 116 48 97)(30 115 49 96)(31 114 50 95)(32 113 51 94)(33 112 52 93)(34 111 53 92)(35 110 54 91)(36 109 46 99)(37 108 61 84)(38 107 62 83)(39 106 63 82)(40 105 55 90)(41 104 56 89)(42 103 57 88)(43 102 58 87)(44 101 59 86)(45 100 60 85)
G:=sub<Sym(144)| (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,49,52)(48,54,51)(56,62,59)(57,60,63)(65,71,68)(66,69,72)(73,79,76)(74,77,80)(82,88,85)(83,86,89)(91,94,97)(93,99,96)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(136,142,139)(137,140,143), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,87,131,102)(2,109,132,99)(3,77,133,123)(4,90,134,105)(5,112,135,93)(6,80,127,126)(7,84,128,108)(8,115,129,96)(9,74,130,120)(10,63,143,39)(11,67,144,21)(12,35,136,54)(13,57,137,42)(14,70,138,24)(15,29,139,48)(16,60,140,45)(17,64,141,27)(18,32,142,51)(19,62,65,38)(20,52,66,33)(22,56,68,41)(23,46,69,36)(25,59,71,44)(26,49,72,30)(28,58,47,43)(31,61,50,37)(34,55,53,40)(73,85,119,100)(75,98,121,117)(76,88,122,103)(78,92,124,111)(79,82,125,106)(81,95,118,114)(83,113,107,94)(86,116,101,97)(89,110,104,91), (1,75,131,121)(2,88,132,103)(3,110,133,91)(4,78,134,124)(5,82,135,106)(6,113,127,94)(7,81,128,118)(8,85,129,100)(9,116,130,97)(10,33,143,52)(11,55,144,40)(12,68,136,22)(13,36,137,46)(14,58,138,43)(15,71,139,25)(16,30,140,49)(17,61,141,37)(18,65,142,19)(20,63,66,39)(21,53,67,34)(23,57,69,42)(24,47,70,28)(26,60,72,45)(27,50,64,31)(29,59,48,44)(32,62,51,38)(35,56,54,41)(73,96,119,115)(74,86,120,101)(76,99,122,109)(77,89,123,104)(79,93,125,112)(80,83,126,107)(84,114,108,95)(87,117,102,98)(90,111,105,92), (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), (1,14,131,138)(2,13,132,137)(3,12,133,136)(4,11,134,144)(5,10,135,143)(6,18,127,142)(7,17,128,141)(8,16,129,140)(9,15,130,139)(19,126,65,80)(20,125,66,79)(21,124,67,78)(22,123,68,77)(23,122,69,76)(24,121,70,75)(25,120,71,74)(26,119,72,73)(27,118,64,81)(28,117,47,98)(29,116,48,97)(30,115,49,96)(31,114,50,95)(32,113,51,94)(33,112,52,93)(34,111,53,92)(35,110,54,91)(36,109,46,99)(37,108,61,84)(38,107,62,83)(39,106,63,82)(40,105,55,90)(41,104,56,89)(42,103,57,88)(43,102,58,87)(44,101,59,86)(45,100,60,85)>;
G:=Group( (2,8,5)(3,6,9)(10,13,16)(12,18,15)(19,25,22)(20,23,26)(29,35,32)(30,33,36)(38,44,41)(39,42,45)(46,49,52)(48,54,51)(56,62,59)(57,60,63)(65,71,68)(66,69,72)(73,79,76)(74,77,80)(82,88,85)(83,86,89)(91,94,97)(93,99,96)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(119,125,122)(120,123,126)(127,130,133)(129,135,132)(136,142,139)(137,140,143), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)(28,34,31)(29,35,32)(30,36,33)(37,43,40)(38,44,41)(39,45,42)(46,52,49)(47,53,50)(48,54,51)(55,61,58)(56,62,59)(57,63,60)(64,70,67)(65,71,68)(66,72,69)(73,79,76)(74,80,77)(75,81,78)(82,88,85)(83,89,86)(84,90,87)(91,97,94)(92,98,95)(93,99,96)(100,106,103)(101,107,104)(102,108,105)(109,115,112)(110,116,113)(111,117,114)(118,124,121)(119,125,122)(120,126,123)(127,133,130)(128,134,131)(129,135,132)(136,142,139)(137,143,140)(138,144,141), (1,87,131,102)(2,109,132,99)(3,77,133,123)(4,90,134,105)(5,112,135,93)(6,80,127,126)(7,84,128,108)(8,115,129,96)(9,74,130,120)(10,63,143,39)(11,67,144,21)(12,35,136,54)(13,57,137,42)(14,70,138,24)(15,29,139,48)(16,60,140,45)(17,64,141,27)(18,32,142,51)(19,62,65,38)(20,52,66,33)(22,56,68,41)(23,46,69,36)(25,59,71,44)(26,49,72,30)(28,58,47,43)(31,61,50,37)(34,55,53,40)(73,85,119,100)(75,98,121,117)(76,88,122,103)(78,92,124,111)(79,82,125,106)(81,95,118,114)(83,113,107,94)(86,116,101,97)(89,110,104,91), (1,75,131,121)(2,88,132,103)(3,110,133,91)(4,78,134,124)(5,82,135,106)(6,113,127,94)(7,81,128,118)(8,85,129,100)(9,116,130,97)(10,33,143,52)(11,55,144,40)(12,68,136,22)(13,36,137,46)(14,58,138,43)(15,71,139,25)(16,30,140,49)(17,61,141,37)(18,65,142,19)(20,63,66,39)(21,53,67,34)(23,57,69,42)(24,47,70,28)(26,60,72,45)(27,50,64,31)(29,59,48,44)(32,62,51,38)(35,56,54,41)(73,96,119,115)(74,86,120,101)(76,99,122,109)(77,89,123,104)(79,93,125,112)(80,83,126,107)(84,114,108,95)(87,117,102,98)(90,111,105,92), (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), (1,14,131,138)(2,13,132,137)(3,12,133,136)(4,11,134,144)(5,10,135,143)(6,18,127,142)(7,17,128,141)(8,16,129,140)(9,15,130,139)(19,126,65,80)(20,125,66,79)(21,124,67,78)(22,123,68,77)(23,122,69,76)(24,121,70,75)(25,120,71,74)(26,119,72,73)(27,118,64,81)(28,117,47,98)(29,116,48,97)(30,115,49,96)(31,114,50,95)(32,113,51,94)(33,112,52,93)(34,111,53,92)(35,110,54,91)(36,109,46,99)(37,108,61,84)(38,107,62,83)(39,106,63,82)(40,105,55,90)(41,104,56,89)(42,103,57,88)(43,102,58,87)(44,101,59,86)(45,100,60,85) );
G=PermutationGroup([[(2,8,5),(3,6,9),(10,13,16),(12,18,15),(19,25,22),(20,23,26),(29,35,32),(30,33,36),(38,44,41),(39,42,45),(46,49,52),(48,54,51),(56,62,59),(57,60,63),(65,71,68),(66,69,72),(73,79,76),(74,77,80),(82,88,85),(83,86,89),(91,94,97),(93,99,96),(100,106,103),(101,104,107),(109,115,112),(110,113,116),(119,125,122),(120,123,126),(127,130,133),(129,135,132),(136,142,139),(137,140,143)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,25,22),(20,26,23),(21,27,24),(28,34,31),(29,35,32),(30,36,33),(37,43,40),(38,44,41),(39,45,42),(46,52,49),(47,53,50),(48,54,51),(55,61,58),(56,62,59),(57,63,60),(64,70,67),(65,71,68),(66,72,69),(73,79,76),(74,80,77),(75,81,78),(82,88,85),(83,89,86),(84,90,87),(91,97,94),(92,98,95),(93,99,96),(100,106,103),(101,107,104),(102,108,105),(109,115,112),(110,116,113),(111,117,114),(118,124,121),(119,125,122),(120,126,123),(127,133,130),(128,134,131),(129,135,132),(136,142,139),(137,143,140),(138,144,141)], [(1,87,131,102),(2,109,132,99),(3,77,133,123),(4,90,134,105),(5,112,135,93),(6,80,127,126),(7,84,128,108),(8,115,129,96),(9,74,130,120),(10,63,143,39),(11,67,144,21),(12,35,136,54),(13,57,137,42),(14,70,138,24),(15,29,139,48),(16,60,140,45),(17,64,141,27),(18,32,142,51),(19,62,65,38),(20,52,66,33),(22,56,68,41),(23,46,69,36),(25,59,71,44),(26,49,72,30),(28,58,47,43),(31,61,50,37),(34,55,53,40),(73,85,119,100),(75,98,121,117),(76,88,122,103),(78,92,124,111),(79,82,125,106),(81,95,118,114),(83,113,107,94),(86,116,101,97),(89,110,104,91)], [(1,75,131,121),(2,88,132,103),(3,110,133,91),(4,78,134,124),(5,82,135,106),(6,113,127,94),(7,81,128,118),(8,85,129,100),(9,116,130,97),(10,33,143,52),(11,55,144,40),(12,68,136,22),(13,36,137,46),(14,58,138,43),(15,71,139,25),(16,30,140,49),(17,61,141,37),(18,65,142,19),(20,63,66,39),(21,53,67,34),(23,57,69,42),(24,47,70,28),(26,60,72,45),(27,50,64,31),(29,59,48,44),(32,62,51,38),(35,56,54,41),(73,96,119,115),(74,86,120,101),(76,99,122,109),(77,89,123,104),(79,93,125,112),(80,83,126,107),(84,114,108,95),(87,117,102,98),(90,111,105,92)], [(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)], [(1,14,131,138),(2,13,132,137),(3,12,133,136),(4,11,134,144),(5,10,135,143),(6,18,127,142),(7,17,128,141),(8,16,129,140),(9,15,130,139),(19,126,65,80),(20,125,66,79),(21,124,67,78),(22,123,68,77),(23,122,69,76),(24,121,70,75),(25,120,71,74),(26,119,72,73),(27,118,64,81),(28,117,47,98),(29,116,48,97),(30,115,49,96),(31,114,50,95),(32,113,51,94),(33,112,52,93),(34,111,53,92),(35,110,54,91),(36,109,46,99),(37,108,61,84),(38,107,62,83),(39,106,63,82),(40,105,55,90),(41,104,56,89),(42,103,57,88),(43,102,58,87),(44,101,59,86),(45,100,60,85)]])
Matrix representation of C32.CSU2(𝔽3) ►in GL8(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 5 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 22 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 | 0 | 0 | 1 |
| 0 | 0 | 5 | 52 | 0 | 0 | 72 | 72 |
| 52 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 21 | 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 | 1 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 |
| 16 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 36 | 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 33 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 24 | 0 | 0 | 22 | 0 | 1 |
| 0 | 0 | 29 | 0 | 0 | 22 | 72 | 72 |
| 0 | 0 | 49 | 0 | 0 | 52 | 0 | 0 |
| 0 | 0 | 49 | 1 | 0 | 52 | 0 | 0 |
| 44 | 45 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 29 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 7 | 0 | 0 | 3 | 45 |
| 0 | 0 | 72 | 7 | 0 | 0 | 42 | 70 |
| 0 | 0 | 48 | 63 | 3 | 45 | 0 | 0 |
| 0 | 0 | 1 | 63 | 42 | 70 | 0 | 0 |
G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,33,0,0,5,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,35,0,33,0,5,0,0,2,71,22,22,52,52,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[52,19,0,0,0,0,0,0,19,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,36,0,0,0,0,0,0,35,56,0,0,0,0,0,0,0,0,51,33,24,29,49,49,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,2,71,22,22,52,52,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0],[44,4,0,0,0,0,0,0,45,29,0,0,0,0,0,0,0,0,39,10,57,72,48,1,0,0,67,34,7,7,63,63,0,0,0,0,0,0,3,42,0,0,0,0,0,0,45,70,0,0,0,0,3,42,0,0,0,0,0,0,45,70,0,0] >;
C32.CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2.{\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2.CSU(2,3)"); // GroupNames label
G:=SmallGroup(432,243);
// by ID
G=gap.SmallGroup(432,243);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,632,261,142,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=1,d^2=f^2=c^2,e^3=f*b*f^-1=b^-1,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*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=b*e^2>;
// generators/relations
Export
Subgroup lattice of C32.CSU2(𝔽3) in TeX
Character table of C32.CSU2(𝔽3) in TeX