G = C32⋊CSU2(𝔽3) order 432 = 24·33
1st semidirect product of C32 and CSU2(𝔽3) acting via CSU2(𝔽3)/Q8=S3
Aliases: C32⋊1CSU2(𝔽3), C6.5S4⋊C3, C6.5(C3×S4), (C3×C6).3S4, Q8⋊He3.1C2, Q8.(C32⋊C6), (Q8×C32).9S3, (C3×SL2(𝔽3)).C6, C2.2(C62⋊S3), C3.3(C3×CSU2(𝔽3)), (C3×Q8).5(C3×S3), SmallGroup(432,247)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C32⋊CSU2(𝔽3) |
| C3×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, eae-1=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=fdf-1=c-1, ece-1=cd, fcf-1=c2d, ede-1=c, fef-1=e-1 >
3C3
12C3
24C3
3C4
18C4
3C6
12C6
24C6
4C32
8C32
9C8
9Q8
3C12
3C12
6Dic3
6C12
18C12
36Dic3
4He3
9Q16
3Dic6
9C24
Character table of C32⋊CSU2(𝔽3)
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 3 | 3 | 24 | 24 | 24 | 6 | 36 | 2 | 3 | 3 | 24 | 24 | 24 | 18 | 18 | 6 | 6 | 12 | 12 | 12 | 36 | 36 | 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 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | 1 | ζ32 | 1 | -1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | -1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | 1 | ζ3 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -√2 | √2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 1 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | √2 | -√2 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ10 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | -1 | ζ65 | ζ6 | 0 | 0 | -1+√-3 | -1-√-3 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ11 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | -1 | ζ6 | ζ65 | 0 | 0 | -1-√-3 | -1+√-3 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×S3 |
| ρ12 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | 1 | ζ3 | ζ32 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | complex lifted from C3×CSU2(𝔽3) |
| ρ13 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | 1 | ζ32 | ζ3 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ32+ζ8ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ3+ζ85ζ3 | complex lifted from C3×CSU2(𝔽3) |
| ρ14 | 2 | -2 | 2 | -1+√-3 | -1-√-3 | ζ65 | -1 | ζ6 | 0 | 0 | -2 | 1+√-3 | 1-√-3 | 1 | ζ32 | ζ3 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | complex lifted from C3×CSU2(𝔽3) |
| ρ15 | 2 | -2 | 2 | -1-√-3 | -1+√-3 | ζ6 | -1 | ζ65 | 0 | 0 | -2 | 1-√-3 | 1+√-3 | 1 | ζ3 | ζ32 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ3+ζ8ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ32+ζ85ζ32 | complex lifted from C3×CSU2(𝔽3) |
| ρ16 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | 1 | 3 | 3 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3×S4 |
| ρ19 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3×S4 |
| ρ20 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | ζ65 | ζ6 | ζ3 | ζ3 | ζ32 | ζ32 | complex lifted from C3×S4 |
| ρ21 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | ζ6 | ζ65 | ζ32 | ζ32 | ζ3 | ζ3 | complex lifted from C3×S4 |
| ρ22 | 4 | -4 | 4 | 4 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -4 | -4 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from CSU2(𝔽3), Schur index 2 |
| ρ23 | 4 | -4 | 4 | -2+2√-3 | -2-2√-3 | ζ3 | 1 | ζ32 | 0 | 0 | -4 | 2+2√-3 | 2-2√-3 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×CSU2(𝔽3) |
| ρ24 | 4 | -4 | 4 | -2-2√-3 | -2+2√-3 | ζ32 | 1 | ζ3 | 0 | 0 | -4 | 2-2√-3 | 2+2√-3 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3×CSU2(𝔽3) |
| ρ25 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ26 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C62⋊S3 |
| ρ27 | 6 | 6 | -3 | 0 | 0 | 0 | 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 | complex lifted from C62⋊S3 |
| ρ28 | 6 | 6 | -3 | 0 | 0 | 0 | 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 | complex lifted from C62⋊S3 |
| ρ29 | 12 | -12 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 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
(5 138 130)(6 139 131)(7 140 132)(8 137 129)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(33 41 50)(34 42 51)(35 43 52)(36 44 49)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(101 109 117)(102 110 118)(103 111 119)(104 112 120)(105 113 122)(106 114 123)(107 115 124)(108 116 121)(125 141 133)(126 142 134)(127 143 135)(128 144 136)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 130 138)(6 131 139)(7 132 140)(8 129 137)(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 50 41)(34 51 42)(35 52 43)(36 49 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(57 76 65)(58 73 66)(59 74 67)(60 75 68)(77 85 93)(78 86 94)(79 87 95)(80 88 96)(81 89 99)(82 90 100)(83 91 97)(84 92 98)(101 109 117)(102 110 118)(103 111 119)(104 112 120)(105 113 122)(106 114 123)(107 115 124)(108 116 121)(125 133 141)(126 134 142)(127 135 143)(128 136 144)
(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 11 3 9)(2 10 4 12)(5 143 7 141)(6 142 8 144)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 42 39 44)(38 41 40 43)(45 51 47 49)(46 50 48 52)(53 58 55 60)(54 57 56 59)(61 66 63 68)(62 65 64 67)(69 73 71 75)(70 76 72 74)(77 83 79 81)(78 82 80 84)(85 91 87 89)(86 90 88 92)(93 97 95 99)(94 100 96 98)(101 106 103 108)(102 105 104 107)(109 114 111 116)(110 113 112 115)(117 123 119 121)(118 122 120 124)(125 130 127 132)(126 129 128 131)(133 138 135 140)(134 137 136 139)
(1 58 29)(2 55 33)(3 60 31)(4 53 35)(5 96 120)(6 98 123)(7 94 118)(8 100 121)(9 56 32)(10 59 34)(11 54 30)(12 57 36)(13 63 41)(14 68 39)(15 61 43)(16 66 37)(17 64 40)(18 67 42)(19 62 38)(20 65 44)(21 71 50)(22 75 47)(23 69 52)(24 73 45)(25 72 48)(26 74 51)(27 70 46)(28 76 49)(77 107 126)(78 102 132)(79 105 128)(80 104 130)(81 103 125)(82 108 129)(83 101 127)(84 106 131)(85 115 134)(86 110 140)(87 113 136)(88 112 138)(89 111 133)(90 116 137)(91 109 135)(92 114 139)(93 124 142)(95 122 144)(97 117 143)(99 119 141)
(1 77 3 79)(2 81 4 83)(5 72 7 70)(6 76 8 74)(9 78 11 80)(10 84 12 82)(13 89 15 91)(14 87 16 85)(17 86 19 88)(18 92 20 90)(21 99 23 97)(22 95 24 93)(25 94 27 96)(26 98 28 100)(29 107 31 105)(30 104 32 102)(33 103 35 101)(34 106 36 108)(37 115 39 113)(38 112 40 110)(41 111 43 109)(42 114 44 116)(45 124 47 122)(46 120 48 118)(49 121 51 123)(50 119 52 117)(53 127 55 125)(54 130 56 132)(57 129 59 131)(58 126 60 128)(61 135 63 133)(62 138 64 140)(65 137 67 139)(66 134 68 136)(69 143 71 141)(73 142 75 144)
G:=sub<Sym(144)| (5,138,130)(6,139,131)(7,140,132)(8,137,129)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(33,41,50)(34,42,51)(35,43,52)(36,44,49)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(105,113,122)(106,114,123)(107,115,124)(108,116,121)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,130,138)(6,131,139)(7,132,140)(8,129,137)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(77,85,93)(78,86,94)(79,87,95)(80,88,96)(81,89,99)(82,90,100)(83,91,97)(84,92,98)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(105,113,122)(106,114,123)(107,115,124)(108,116,121)(125,133,141)(126,134,142)(127,135,143)(128,136,144), (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), (1,11,3,9)(2,10,4,12)(5,143,7,141)(6,142,8,144)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43)(45,51,47,49)(46,50,48,52)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67)(69,73,71,75)(70,76,72,74)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92)(93,97,95,99)(94,100,96,98)(101,106,103,108)(102,105,104,107)(109,114,111,116)(110,113,112,115)(117,123,119,121)(118,122,120,124)(125,130,127,132)(126,129,128,131)(133,138,135,140)(134,137,136,139), (1,58,29)(2,55,33)(3,60,31)(4,53,35)(5,96,120)(6,98,123)(7,94,118)(8,100,121)(9,56,32)(10,59,34)(11,54,30)(12,57,36)(13,63,41)(14,68,39)(15,61,43)(16,66,37)(17,64,40)(18,67,42)(19,62,38)(20,65,44)(21,71,50)(22,75,47)(23,69,52)(24,73,45)(25,72,48)(26,74,51)(27,70,46)(28,76,49)(77,107,126)(78,102,132)(79,105,128)(80,104,130)(81,103,125)(82,108,129)(83,101,127)(84,106,131)(85,115,134)(86,110,140)(87,113,136)(88,112,138)(89,111,133)(90,116,137)(91,109,135)(92,114,139)(93,124,142)(95,122,144)(97,117,143)(99,119,141), (1,77,3,79)(2,81,4,83)(5,72,7,70)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,89,15,91)(14,87,16,85)(17,86,19,88)(18,92,20,90)(21,99,23,97)(22,95,24,93)(25,94,27,96)(26,98,28,100)(29,107,31,105)(30,104,32,102)(33,103,35,101)(34,106,36,108)(37,115,39,113)(38,112,40,110)(41,111,43,109)(42,114,44,116)(45,124,47,122)(46,120,48,118)(49,121,51,123)(50,119,52,117)(53,127,55,125)(54,130,56,132)(57,129,59,131)(58,126,60,128)(61,135,63,133)(62,138,64,140)(65,137,67,139)(66,134,68,136)(69,143,71,141)(73,142,75,144)>;
G:=Group( (5,138,130)(6,139,131)(7,140,132)(8,137,129)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(33,41,50)(34,42,51)(35,43,52)(36,44,49)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(105,113,122)(106,114,123)(107,115,124)(108,116,121)(125,141,133)(126,142,134)(127,143,135)(128,144,136), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,130,138)(6,131,139)(7,132,140)(8,129,137)(9,25,17)(10,26,18)(11,27,19)(12,28,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(57,76,65)(58,73,66)(59,74,67)(60,75,68)(77,85,93)(78,86,94)(79,87,95)(80,88,96)(81,89,99)(82,90,100)(83,91,97)(84,92,98)(101,109,117)(102,110,118)(103,111,119)(104,112,120)(105,113,122)(106,114,123)(107,115,124)(108,116,121)(125,133,141)(126,134,142)(127,135,143)(128,136,144), (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), (1,11,3,9)(2,10,4,12)(5,143,7,141)(6,142,8,144)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,42,39,44)(38,41,40,43)(45,51,47,49)(46,50,48,52)(53,58,55,60)(54,57,56,59)(61,66,63,68)(62,65,64,67)(69,73,71,75)(70,76,72,74)(77,83,79,81)(78,82,80,84)(85,91,87,89)(86,90,88,92)(93,97,95,99)(94,100,96,98)(101,106,103,108)(102,105,104,107)(109,114,111,116)(110,113,112,115)(117,123,119,121)(118,122,120,124)(125,130,127,132)(126,129,128,131)(133,138,135,140)(134,137,136,139), (1,58,29)(2,55,33)(3,60,31)(4,53,35)(5,96,120)(6,98,123)(7,94,118)(8,100,121)(9,56,32)(10,59,34)(11,54,30)(12,57,36)(13,63,41)(14,68,39)(15,61,43)(16,66,37)(17,64,40)(18,67,42)(19,62,38)(20,65,44)(21,71,50)(22,75,47)(23,69,52)(24,73,45)(25,72,48)(26,74,51)(27,70,46)(28,76,49)(77,107,126)(78,102,132)(79,105,128)(80,104,130)(81,103,125)(82,108,129)(83,101,127)(84,106,131)(85,115,134)(86,110,140)(87,113,136)(88,112,138)(89,111,133)(90,116,137)(91,109,135)(92,114,139)(93,124,142)(95,122,144)(97,117,143)(99,119,141), (1,77,3,79)(2,81,4,83)(5,72,7,70)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,89,15,91)(14,87,16,85)(17,86,19,88)(18,92,20,90)(21,99,23,97)(22,95,24,93)(25,94,27,96)(26,98,28,100)(29,107,31,105)(30,104,32,102)(33,103,35,101)(34,106,36,108)(37,115,39,113)(38,112,40,110)(41,111,43,109)(42,114,44,116)(45,124,47,122)(46,120,48,118)(49,121,51,123)(50,119,52,117)(53,127,55,125)(54,130,56,132)(57,129,59,131)(58,126,60,128)(61,135,63,133)(62,138,64,140)(65,137,67,139)(66,134,68,136)(69,143,71,141)(73,142,75,144) );
G=PermutationGroup([[(5,138,130),(6,139,131),(7,140,132),(8,137,129),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(33,41,50),(34,42,51),(35,43,52),(36,44,49),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,76,65),(58,73,66),(59,74,67),(60,75,68),(101,109,117),(102,110,118),(103,111,119),(104,112,120),(105,113,122),(106,114,123),(107,115,124),(108,116,121),(125,141,133),(126,142,134),(127,143,135),(128,144,136)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,130,138),(6,131,139),(7,132,140),(8,129,137),(9,25,17),(10,26,18),(11,27,19),(12,28,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,50,41),(34,51,42),(35,52,43),(36,49,44),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(57,76,65),(58,73,66),(59,74,67),(60,75,68),(77,85,93),(78,86,94),(79,87,95),(80,88,96),(81,89,99),(82,90,100),(83,91,97),(84,92,98),(101,109,117),(102,110,118),(103,111,119),(104,112,120),(105,113,122),(106,114,123),(107,115,124),(108,116,121),(125,133,141),(126,134,142),(127,135,143),(128,136,144)], [(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)], [(1,11,3,9),(2,10,4,12),(5,143,7,141),(6,142,8,144),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27),(29,34,31,36),(30,33,32,35),(37,42,39,44),(38,41,40,43),(45,51,47,49),(46,50,48,52),(53,58,55,60),(54,57,56,59),(61,66,63,68),(62,65,64,67),(69,73,71,75),(70,76,72,74),(77,83,79,81),(78,82,80,84),(85,91,87,89),(86,90,88,92),(93,97,95,99),(94,100,96,98),(101,106,103,108),(102,105,104,107),(109,114,111,116),(110,113,112,115),(117,123,119,121),(118,122,120,124),(125,130,127,132),(126,129,128,131),(133,138,135,140),(134,137,136,139)], [(1,58,29),(2,55,33),(3,60,31),(4,53,35),(5,96,120),(6,98,123),(7,94,118),(8,100,121),(9,56,32),(10,59,34),(11,54,30),(12,57,36),(13,63,41),(14,68,39),(15,61,43),(16,66,37),(17,64,40),(18,67,42),(19,62,38),(20,65,44),(21,71,50),(22,75,47),(23,69,52),(24,73,45),(25,72,48),(26,74,51),(27,70,46),(28,76,49),(77,107,126),(78,102,132),(79,105,128),(80,104,130),(81,103,125),(82,108,129),(83,101,127),(84,106,131),(85,115,134),(86,110,140),(87,113,136),(88,112,138),(89,111,133),(90,116,137),(91,109,135),(92,114,139),(93,124,142),(95,122,144),(97,117,143),(99,119,141)], [(1,77,3,79),(2,81,4,83),(5,72,7,70),(6,76,8,74),(9,78,11,80),(10,84,12,82),(13,89,15,91),(14,87,16,85),(17,86,19,88),(18,92,20,90),(21,99,23,97),(22,95,24,93),(25,94,27,96),(26,98,28,100),(29,107,31,105),(30,104,32,102),(33,103,35,101),(34,106,36,108),(37,115,39,113),(38,112,40,110),(41,111,43,109),(42,114,44,116),(45,124,47,122),(46,120,48,118),(49,121,51,123),(50,119,52,117),(53,127,55,125),(54,130,56,132),(57,129,59,131),(58,126,60,128),(61,135,63,133),(62,138,64,140),(65,137,67,139),(66,134,68,136),(69,143,71,141),(73,142,75,144)]])
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 | 0 | 0 | 36 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 45 |
| 0 | 0 | 0 | 0 | 0 | 0 | 45 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 36 | 45 |
| 0 | 0 | 0 | 0 | 0 | 0 | 45 | 36 |
| 12 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 61 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 6 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 66 | 0 | 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 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 41 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 53 |
| 0 | 0 | 0 | 0 | 0 | 0 | 20 | 67 |
| 0 | 0 | 0 | 0 | 6 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 67 | 0 | 0 |
G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,28,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,36,45,0,0,0,0,0,0,45,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,45,0,0,0,0,0,0,45,36,0,0,0,0,0,0,0,0,36,45,0,0,0,0,0,0,45,36,0,0,0,0,0,0,0,0,36,45,0,0,0,0,0,0,45,36],[12,72,0,0,0,0,0,0,72,61,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0],[6,6,0,0,0,0,0,0,5,66,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[46,41,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,6,20,0,0,0,0,0,0,53,67,0,0,0,0,0,0,0,0,0,0,6,20,0,0,0,0,0,0,53,67,0,0,0,0,6,20,0,0,0,0,0,0,53,67,0,0] >;
C32⋊CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2\rtimes {\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2:CSU(2,3)"); // GroupNames label
G:=SmallGroup(432,247);
// by ID
G=gap.SmallGroup(432,247);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,254,261,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=e^3=1,d^2=f^2=c^2,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,f*b*f^-1=b^-1,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
Export
Subgroup lattice of C32⋊CSU2(𝔽3) in TeX
Character table of C32⋊CSU2(𝔽3) in TeX