direct product, non-abelian, soluble
Aliases: C2×Q8⋊D9, Q8⋊D18, C6.2GL2(𝔽3), (C2×C6).8S4, (C2×Q8)⋊1D9, C6.20(C2×S4), Q8⋊C9⋊1C22, (C3×Q8).9D6, (C6×Q8).3S3, C3.(C2×GL2(𝔽3)), C22.5(C3.S4), (C2×Q8⋊C9)⋊2C2, C2.6(C2×C3.S4), SmallGroup(288,336)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C2×Q8⋊D9 |
Generators and relations for C2×Q8⋊D9
G = < a,b,c,d,e | a2=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe=b-1c, dcd-1=bc, ece=b2c, ede=d-1 >
Subgroups: 527 in 81 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, C9, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, D9, C18, C3⋊C8, D12, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×SD16, D18, C2×C18, C2×C3⋊C8, Q8⋊2S3, C2×D12, C6×Q8, Q8⋊C9, C22×D9, C2×Q8⋊2S3, Q8⋊D9, C2×Q8⋊C9, C2×Q8⋊D9
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, GL2(𝔽3), C2×S4, C3.S4, C2×GL2(𝔽3), Q8⋊D9, C2×C3.S4, C2×Q8⋊D9
Character table of C2×Q8⋊D9
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 18D | 18E | 18F | 18G | 18H | 18I | |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 6 | 6 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 8 | 8 | 8 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 1 | linear of order 2 |
| ρ3 | 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 | -1 | linear of order 2 |
| ρ4 | 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 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | 1 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | 1 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | 1 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | -√-2 | -√-2 | √-2 | √-2 | -1 | -1 | -1 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | complex lifted from GL2(𝔽3) |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | √-2 | -√-2 | √-2 | -√-2 | -1 | -1 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | √-2 | √-2 | -√-2 | -√-2 | -1 | -1 | -1 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | complex lifted from GL2(𝔽3) |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | -√-2 | √-2 | -√-2 | √-2 | -1 | -1 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | complex lifted from GL2(𝔽3) |
| ρ17 | 3 | 3 | -3 | -3 | -1 | 1 | 3 | -1 | 1 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | 3 | 3 | 1 | 1 | 3 | -1 | -1 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -3 | -3 | 1 | -1 | 3 | -1 | 1 | -3 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | orthogonal lifted from GL2(𝔽3) |
| ρ22 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | orthogonal lifted from GL2(𝔽3) |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 0 | 0 | -ζ98-ζ9 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | -ζ95-ζ94 | ζ98+ζ9 | orthogonal lifted from Q8⋊D9 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 0 | 0 | ζ98+ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | orthogonal lifted from Q8⋊D9 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 0 | 0 | ζ95+ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | orthogonal lifted from Q8⋊D9 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 0 | 0 | ζ97+ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | orthogonal lifted from Q8⋊D9 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 0 | 0 | -ζ95-ζ94 | ζ98+ζ9 | -ζ98-ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | -ζ97-ζ92 | ζ95+ζ94 | orthogonal lifted from Q8⋊D9 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 0 | 0 | -ζ97-ζ92 | ζ95+ζ94 | -ζ95-ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | -ζ98-ζ9 | ζ97+ζ92 | orthogonal lifted from Q8⋊D9 |
| ρ29 | 6 | 6 | -6 | -6 | 0 | 0 | -3 | -2 | 2 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ30 | 6 | 6 | 6 | 6 | 0 | 0 | -3 | -2 | -2 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
►On 144 points
Generators in S144
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 136)(9 137)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 131)(20 132)(21 133)(22 134)(23 135)(24 127)(25 128)(26 129)(27 130)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 82)(45 83)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 100)(71 101)(72 102)
(1 113 77 106)(2 85 78 128)(3 126 79 96)(4 116 80 100)(5 88 81 131)(6 120 73 99)(7 110 74 103)(8 82 75 134)(9 123 76 93)(10 36 143 57)(11 64 144 47)(12 22 136 44)(13 30 137 60)(14 67 138 50)(15 25 139 38)(16 33 140 63)(17 70 141 53)(18 19 142 41)(20 72 42 46)(21 58 43 28)(23 66 45 49)(24 61 37 31)(26 69 39 52)(27 55 40 34)(29 48 59 65)(32 51 62 68)(35 54 56 71)(83 112 135 105)(84 94 127 124)(86 115 129 108)(87 97 130 118)(89 109 132 102)(90 91 133 121)(92 111 122 104)(95 114 125 107)(98 117 119 101)
(1 84 77 127)(2 125 78 95)(3 115 79 108)(4 87 80 130)(5 119 81 98)(6 109 73 102)(7 90 74 133)(8 122 75 92)(9 112 76 105)(10 72 143 46)(11 21 144 43)(12 29 136 59)(13 66 137 49)(14 24 138 37)(15 32 139 62)(16 69 140 52)(17 27 141 40)(18 35 142 56)(19 71 41 54)(20 57 42 36)(22 65 44 48)(23 60 45 30)(25 68 38 51)(26 63 39 33)(28 47 58 64)(31 50 61 67)(34 53 55 70)(82 111 134 104)(83 93 135 123)(85 114 128 107)(86 96 129 126)(88 117 131 101)(89 99 132 120)(91 110 121 103)(94 113 124 106)(97 116 118 100)
(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 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 63)(29 62)(30 61)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 66)(38 65)(39 64)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(73 80)(74 79)(75 78)(76 77)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 108)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 118)(109 130)(110 129)(111 128)(112 127)(113 135)(114 134)(115 133)(116 132)(117 131)(136 139)(137 138)(140 144)(141 143)
G:=sub<Sym(144)| (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,131)(20,132)(21,133)(22,134)(23,135)(24,127)(25,128)(26,129)(27,130)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,82)(45,83)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,100)(71,101)(72,102), (1,113,77,106)(2,85,78,128)(3,126,79,96)(4,116,80,100)(5,88,81,131)(6,120,73,99)(7,110,74,103)(8,82,75,134)(9,123,76,93)(10,36,143,57)(11,64,144,47)(12,22,136,44)(13,30,137,60)(14,67,138,50)(15,25,139,38)(16,33,140,63)(17,70,141,53)(18,19,142,41)(20,72,42,46)(21,58,43,28)(23,66,45,49)(24,61,37,31)(26,69,39,52)(27,55,40,34)(29,48,59,65)(32,51,62,68)(35,54,56,71)(83,112,135,105)(84,94,127,124)(86,115,129,108)(87,97,130,118)(89,109,132,102)(90,91,133,121)(92,111,122,104)(95,114,125,107)(98,117,119,101), (1,84,77,127)(2,125,78,95)(3,115,79,108)(4,87,80,130)(5,119,81,98)(6,109,73,102)(7,90,74,133)(8,122,75,92)(9,112,76,105)(10,72,143,46)(11,21,144,43)(12,29,136,59)(13,66,137,49)(14,24,138,37)(15,32,139,62)(16,69,140,52)(17,27,141,40)(18,35,142,56)(19,71,41,54)(20,57,42,36)(22,65,44,48)(23,60,45,30)(25,68,38,51)(26,63,39,33)(28,47,58,64)(31,50,61,67)(34,53,55,70)(82,111,134,104)(83,93,135,123)(85,114,128,107)(86,96,129,126)(88,117,131,101)(89,99,132,120)(91,110,121,103)(94,113,124,106)(97,116,118,100), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(73,80)(74,79)(75,78)(76,77)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,108)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(109,130)(110,129)(111,128)(112,127)(113,135)(114,134)(115,133)(116,132)(117,131)(136,139)(137,138)(140,144)(141,143)>;
G:=Group( (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,136)(9,137)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,131)(20,132)(21,133)(22,134)(23,135)(24,127)(25,128)(26,129)(27,130)(28,91)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,84)(38,85)(39,86)(40,87)(41,88)(42,89)(43,90)(44,82)(45,83)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,120)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,100)(71,101)(72,102), (1,113,77,106)(2,85,78,128)(3,126,79,96)(4,116,80,100)(5,88,81,131)(6,120,73,99)(7,110,74,103)(8,82,75,134)(9,123,76,93)(10,36,143,57)(11,64,144,47)(12,22,136,44)(13,30,137,60)(14,67,138,50)(15,25,139,38)(16,33,140,63)(17,70,141,53)(18,19,142,41)(20,72,42,46)(21,58,43,28)(23,66,45,49)(24,61,37,31)(26,69,39,52)(27,55,40,34)(29,48,59,65)(32,51,62,68)(35,54,56,71)(83,112,135,105)(84,94,127,124)(86,115,129,108)(87,97,130,118)(89,109,132,102)(90,91,133,121)(92,111,122,104)(95,114,125,107)(98,117,119,101), (1,84,77,127)(2,125,78,95)(3,115,79,108)(4,87,80,130)(5,119,81,98)(6,109,73,102)(7,90,74,133)(8,122,75,92)(9,112,76,105)(10,72,143,46)(11,21,144,43)(12,29,136,59)(13,66,137,49)(14,24,138,37)(15,32,139,62)(16,69,140,52)(17,27,141,40)(18,35,142,56)(19,71,41,54)(20,57,42,36)(22,65,44,48)(23,60,45,30)(25,68,38,51)(26,63,39,33)(28,47,58,64)(31,50,61,67)(34,53,55,70)(82,111,134,104)(83,93,135,123)(85,114,128,107)(86,96,129,126)(88,117,131,101)(89,99,132,120)(91,110,121,103)(94,113,124,106)(97,116,118,100), (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,66)(38,65)(39,64)(40,72)(41,71)(42,70)(43,69)(44,68)(45,67)(73,80)(74,79)(75,78)(76,77)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,108)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,118)(109,130)(110,129)(111,128)(112,127)(113,135)(114,134)(115,133)(116,132)(117,131)(136,139)(137,138)(140,144)(141,143) );
G=PermutationGroup([[(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,136),(9,137),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,131),(20,132),(21,133),(22,134),(23,135),(24,127),(25,128),(26,129),(27,130),(28,91),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,84),(38,85),(39,86),(40,87),(41,88),(42,89),(43,90),(44,82),(45,83),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,120),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,100),(71,101),(72,102)], [(1,113,77,106),(2,85,78,128),(3,126,79,96),(4,116,80,100),(5,88,81,131),(6,120,73,99),(7,110,74,103),(8,82,75,134),(9,123,76,93),(10,36,143,57),(11,64,144,47),(12,22,136,44),(13,30,137,60),(14,67,138,50),(15,25,139,38),(16,33,140,63),(17,70,141,53),(18,19,142,41),(20,72,42,46),(21,58,43,28),(23,66,45,49),(24,61,37,31),(26,69,39,52),(27,55,40,34),(29,48,59,65),(32,51,62,68),(35,54,56,71),(83,112,135,105),(84,94,127,124),(86,115,129,108),(87,97,130,118),(89,109,132,102),(90,91,133,121),(92,111,122,104),(95,114,125,107),(98,117,119,101)], [(1,84,77,127),(2,125,78,95),(3,115,79,108),(4,87,80,130),(5,119,81,98),(6,109,73,102),(7,90,74,133),(8,122,75,92),(9,112,76,105),(10,72,143,46),(11,21,144,43),(12,29,136,59),(13,66,137,49),(14,24,138,37),(15,32,139,62),(16,69,140,52),(17,27,141,40),(18,35,142,56),(19,71,41,54),(20,57,42,36),(22,65,44,48),(23,60,45,30),(25,68,38,51),(26,63,39,33),(28,47,58,64),(31,50,61,67),(34,53,55,70),(82,111,134,104),(83,93,135,123),(85,114,128,107),(86,96,129,126),(88,117,131,101),(89,99,132,120),(91,110,121,103),(94,113,124,106),(97,116,118,100)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,63),(29,62),(30,61),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,66),(38,65),(39,64),(40,72),(41,71),(42,70),(43,69),(44,68),(45,67),(73,80),(74,79),(75,78),(76,77),(82,107),(83,106),(84,105),(85,104),(86,103),(87,102),(88,101),(89,100),(90,108),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,118),(109,130),(110,129),(111,128),(112,127),(113,135),(114,134),(115,133),(116,132),(117,131),(136,139),(137,138),(140,144),(141,143)]])
Matrix representation of C2×Q8⋊D9 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 72 | 0 | 0 |
| 0 | 72 | 61 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 66 | 68 | 0 | 0 |
| 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 0 | 28 | 3 |
| 0 | 0 | 0 | 70 | 31 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 66 | 68 | 0 | 0 |
| 0 | 68 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 72 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,72,0,0,0,72,61,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,66,67,0,0,0,68,6,0,0,0,0,0,28,70,0,0,0,3,31],[72,0,0,0,0,0,66,68,0,0,0,68,7,0,0,0,0,0,1,1,0,0,0,0,72] >;
C2×Q8⋊D9 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes D_9
% in TeX
G:=Group("C2xQ8:D9"); // GroupNames label
G:=SmallGroup(288,336);
// by ID
G=gap.SmallGroup(288,336);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^9=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=c,e*b*e=b^-1*c,d*c*d^-1=b*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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