G = C32⋊2CSU2(𝔽3) order 432 = 24·33
2nd semidirect product of C32 and CSU2(𝔽3) acting via CSU2(𝔽3)/Q8=S3
Aliases: C32⋊2CSU2(𝔽3), (C3×C6).7S4, Q8⋊He3.2C2, C6.17(C3⋊S4), Q8.(He3⋊C2), C3.3(C6.5S4), (Q8×C32).12S3, C2.2(C32⋊S4), (C3×SL2(𝔽3)).3S3, (C3×Q8).6(C3⋊S3), SmallGroup(432,257)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊He3 — C32⋊2CSU2(𝔽3) |
Generators and relations for C32⋊2CSU2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=1, d2=f2=c2, ab=ba, ac=ca, ad=da, eae-1=ab-1, faf-1=a-1, 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: 368 in 68 conjugacy classes, 14 normal (11 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C3⋊C8, C24, SL2(𝔽3), Dic6, C3×Q8, C3×Q8, He3, C3×Dic3, C3×C12, C3⋊Q16, C3×Q16, CSU2(𝔽3), C2×He3, C3×C3⋊C8, C3×SL2(𝔽3), C3×Dic6, Q8×C32, He3⋊3C4, C3×C3⋊Q16, C3×CSU2(𝔽3), Q8⋊He3, C32⋊2CSU2(𝔽3)
Quotients: C1, C2, S3, C3⋊S3, S4, CSU2(𝔽3), He3⋊C2, C3⋊S4, C6.5S4, C32⋊S4, C32⋊2CSU2(𝔽3)
Character table of C32⋊2CSU2(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 | 1 | 1 | 6 | 24 | 24 | 24 | 6 | 36 | 1 | 1 | 6 | 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 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 0 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 2 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 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 |
| ρ7 | 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 |
| 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 | 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 |
| ρ10 | 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 |
| ρ11 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from He3⋊C2 |
| ρ12 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from He3⋊C2 |
| ρ13 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from He3⋊C2 |
| ρ14 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | ζ6 | ζ65 | -1-√-3 | -1+√-3 | 2 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from C32⋊S4 |
| ρ15 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | ζ6 | ζ65 | -1-√-3 | -1+√-3 | 2 | ζ65 | ζ6 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from C32⋊S4 |
| ρ16 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | ζ65 | ζ6 | -1+√-3 | -1-√-3 | 2 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from C32⋊S4 |
| ρ17 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | ζ65 | ζ6 | -1+√-3 | -1-√-3 | 2 | ζ6 | ζ65 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from C32⋊S4 |
| ρ18 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 1 | 1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from He3⋊C2 |
| ρ19 | 4 | -4 | 4 | 4 | -2 | -2 | 1 | 1 | 0 | 0 | -4 | -4 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ20 | 4 | -4 | 4 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | -4 | -4 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ21 | 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 |
| ρ22 | 4 | -4 | 4 | 4 | -2 | 1 | 1 | -2 | 0 | 0 | -4 | -4 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C6.5S4, Schur index 2 |
| ρ23 | 6 | 6 | 6 | 6 | -3 | 0 | 0 | 0 | -2 | 0 | 6 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊S4 |
| ρ24 | 6 | 6 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | -2 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1-√-3 | 1+√-3 | 1-√-3 | 1+√-3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊S4 |
| ρ25 | 6 | 6 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | -2 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 1+√-3 | 1-√-3 | 1+√-3 | 1-√-3 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊S4 |
| ρ26 | 6 | -6 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3 | 3-3√-3 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | complex faithful |
| ρ27 | 6 | -6 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 3+3√-3 | 3-3√-3 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ3+ζ85ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ87ζ32+ζ85ζ32 | complex faithful |
| ρ28 | 6 | -6 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3 | 3+3√-3 | 0 | 0 | 0 | 0 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | complex faithful |
| ρ29 | 6 | -6 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 3-3√-3 | 3+3√-3 | 0 | 0 | 0 | 0 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ87ζ32+ζ85ζ32 | -ζ83ζ3+ζ8ζ3 | -ζ83ζ32+ζ8ζ32 | -ζ87ζ3+ζ85ζ3 | complex faithful |
►On 144 points
Generators in S144
(5 138 130)(6 139 131)(7 140 132)(8 137 129)(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 61 69)(54 62 70)(55 63 71)(56 64 72)(57 65 76)(58 66 73)(59 67 74)(60 68 75)(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 22 14)(2 23 15)(3 24 16)(4 21 13)(5 138 130)(6 139 131)(7 140 132)(8 137 129)(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 93 85)(78 94 86)(79 95 87)(80 96 88)(81 99 89)(82 100 90)(83 97 91)(84 98 92)(101 117 109)(102 118 110)(103 119 111)(104 120 112)(105 122 113)(106 123 114)(107 124 115)(108 121 116)(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)(5 141 7 143)(6 144 8 142)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)(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)(125 132 127 130)(126 131 128 129)(133 140 135 138)(134 139 136 137)
(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 61 43)(14 66 37)(15 63 41)(16 68 39)(17 64 40)(18 67 42)(19 62 38)(20 65 44)(21 69 52)(22 73 45)(23 71 50)(24 75 47)(25 72 48)(26 74 51)(27 70 46)(28 76 49)(77 107 128)(78 102 132)(79 105 126)(80 104 130)(81 103 127)(82 108 129)(83 101 125)(84 106 131)(85 115 136)(86 110 140)(87 113 134)(88 112 138)(89 111 135)(90 116 137)(91 109 133)(92 114 139)(93 124 144)(95 122 142)(97 117 141)(99 119 143)
(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 91 15 89)(14 85 16 87)(17 86 19 88)(18 92 20 90)(21 97 23 99)(22 93 24 95)(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)(53 125 55 127)(54 130 56 132)(57 129 59 131)(58 128 60 126)(61 133 63 135)(62 138 64 140)(65 137 67 139)(66 136 68 134)(69 141 71 143)(73 144 75 142)
G:=sub<Sym(144)| (5,138,130)(6,139,131)(7,140,132)(8,137,129)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(57,65,76)(58,66,73)(59,67,74)(60,68,75)(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), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(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)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,99,89)(82,100,90)(83,97,91)(84,98,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(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)(5,141,7,143)(6,144,8,142)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(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)(125,132,127,130)(126,131,128,129)(133,140,135,138)(134,139,136,137), (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)(13,61,43)(14,66,37)(15,63,41)(16,68,39)(17,64,40)(18,67,42)(19,62,38)(20,65,44)(21,69,52)(22,73,45)(23,71,50)(24,75,47)(25,72,48)(26,74,51)(27,70,46)(28,76,49)(77,107,128)(78,102,132)(79,105,126)(80,104,130)(81,103,127)(82,108,129)(83,101,125)(84,106,131)(85,115,136)(86,110,140)(87,113,134)(88,112,138)(89,111,135)(90,116,137)(91,109,133)(92,114,139)(93,124,144)(95,122,142)(97,117,141)(99,119,143), (1,77,3,79)(2,81,4,83)(5,72,7,70)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,91,15,89)(14,85,16,87)(17,86,19,88)(18,92,20,90)(21,97,23,99)(22,93,24,95)(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)(53,125,55,127)(54,130,56,132)(57,129,59,131)(58,128,60,126)(61,133,63,135)(62,138,64,140)(65,137,67,139)(66,136,68,134)(69,141,71,143)(73,144,75,142)>;
G:=Group( (5,138,130)(6,139,131)(7,140,132)(8,137,129)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(33,50,41)(34,51,42)(35,52,43)(36,49,44)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(57,65,76)(58,66,73)(59,67,74)(60,68,75)(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), (1,22,14)(2,23,15)(3,24,16)(4,21,13)(5,138,130)(6,139,131)(7,140,132)(8,137,129)(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)(77,93,85)(78,94,86)(79,95,87)(80,96,88)(81,99,89)(82,100,90)(83,97,91)(84,98,92)(101,117,109)(102,118,110)(103,119,111)(104,120,112)(105,122,113)(106,123,114)(107,124,115)(108,121,116)(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)(5,141,7,143)(6,144,8,142)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)(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)(125,132,127,130)(126,131,128,129)(133,140,135,138)(134,139,136,137), (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)(13,61,43)(14,66,37)(15,63,41)(16,68,39)(17,64,40)(18,67,42)(19,62,38)(20,65,44)(21,69,52)(22,73,45)(23,71,50)(24,75,47)(25,72,48)(26,74,51)(27,70,46)(28,76,49)(77,107,128)(78,102,132)(79,105,126)(80,104,130)(81,103,127)(82,108,129)(83,101,125)(84,106,131)(85,115,136)(86,110,140)(87,113,134)(88,112,138)(89,111,135)(90,116,137)(91,109,133)(92,114,139)(93,124,144)(95,122,142)(97,117,141)(99,119,143), (1,77,3,79)(2,81,4,83)(5,72,7,70)(6,76,8,74)(9,78,11,80)(10,84,12,82)(13,91,15,89)(14,85,16,87)(17,86,19,88)(18,92,20,90)(21,97,23,99)(22,93,24,95)(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)(53,125,55,127)(54,130,56,132)(57,129,59,131)(58,128,60,126)(61,133,63,135)(62,138,64,140)(65,137,67,139)(66,136,68,134)(69,141,71,143)(73,144,75,142) );
G=PermutationGroup([[(5,138,130),(6,139,131),(7,140,132),(8,137,129),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(33,50,41),(34,51,42),(35,52,43),(36,49,44),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(57,65,76),(58,66,73),(59,67,74),(60,68,75),(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)], [(1,22,14),(2,23,15),(3,24,16),(4,21,13),(5,138,130),(6,139,131),(7,140,132),(8,137,129),(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),(77,93,85),(78,94,86),(79,95,87),(80,96,88),(81,99,89),(82,100,90),(83,97,91),(84,98,92),(101,117,109),(102,118,110),(103,119,111),(104,120,112),(105,122,113),(106,123,114),(107,124,115),(108,121,116),(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),(5,141,7,143),(6,144,8,142),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25),(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),(125,132,127,130),(126,131,128,129),(133,140,135,138),(134,139,136,137)], [(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),(13,61,43),(14,66,37),(15,63,41),(16,68,39),(17,64,40),(18,67,42),(19,62,38),(20,65,44),(21,69,52),(22,73,45),(23,71,50),(24,75,47),(25,72,48),(26,74,51),(27,70,46),(28,76,49),(77,107,128),(78,102,132),(79,105,126),(80,104,130),(81,103,127),(82,108,129),(83,101,125),(84,106,131),(85,115,136),(86,110,140),(87,113,134),(88,112,138),(89,111,135),(90,116,137),(91,109,133),(92,114,139),(93,124,144),(95,122,142),(97,117,141),(99,119,143)], [(1,77,3,79),(2,81,4,83),(5,72,7,70),(6,76,8,74),(9,78,11,80),(10,84,12,82),(13,91,15,89),(14,85,16,87),(17,86,19,88),(18,92,20,90),(21,97,23,99),(22,93,24,95),(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),(53,125,55,127),(54,130,56,132),(57,129,59,131),(58,128,60,126),(61,133,63,135),(62,138,64,140),(65,137,67,139),(66,136,68,134),(69,141,71,143),(73,144,75,142)]])
Matrix representation of C32⋊2CSU2(𝔽3) ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 8 | 47 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 72 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 |
| 1 | 61 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 6 | 67 | 0 | 0 | 0 |
| 68 | 66 | 0 | 0 | 0 |
| 0 | 0 | 13 | 1 | 26 |
| 0 | 0 | 19 | 0 | 48 |
| 0 | 0 | 4 | 0 | 60 |
| 27 | 41 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 72 | 60 | 65 |
| 0 | 0 | 0 | 54 | 17 |
| 0 | 0 | 0 | 69 | 19 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,4,47,64],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,1,0,0,0,1,61,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[6,68,0,0,0,67,66,0,0,0,0,0,13,19,4,0,0,1,0,0,0,0,26,48,60],[27,0,0,0,0,41,46,0,0,0,0,0,72,0,0,0,0,60,54,69,0,0,65,17,19] >;
C32⋊2CSU2(𝔽3) in GAP, Magma, Sage, TeX
C_3^2\rtimes_2{\rm CSU}_2({\mathbb F}_3) % in TeX
G:=Group("C3^2:2CSU(2,3)"); // GroupNames label
G:=SmallGroup(432,257);
// by ID
G=gap.SmallGroup(432,257);
# by ID
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,57,254,261,3784,5681,172,2273,3414,285,124]);
// 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,e*a*e^-1=a*b^-1,f*a*f^-1=a^-1,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
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