metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.1F5, Dic20.2C4, D10.5SD16, Dic5.21D8, D5⋊C16⋊2C2, C40.9(C2×C4), (C5×D8).2C4, C8.11(C2×F5), (C4×D5).22D4, C5⋊2C8.13D4, D10.Q8⋊1C2, D8⋊3D5.4C2, C5⋊1(D8.C4), C4.3(C22⋊F5), C20.3(C22⋊C4), C2.8(D20⋊C4), (C8×D5).18C22, C10.7(D4⋊C4), SmallGroup(320,243)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.F5
G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c3 >
8C2
10C2
4C22
5C4
5C22
20C4
2D5
8C10
2D4
5C8
10Q8
20C2×C4
20D4
20C8
4Dic5
5Q16
10C4○D4
10SD16
10M4(2)
10C16
Character table of D8.F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20 | 40A | 40B | |
| size | 1 | 1 | 8 | 10 | 2 | 5 | 5 | 40 | 4 | 2 | 2 | 10 | 10 | 40 | 40 | 4 | 16 | 16 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | 1 | 1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | i | -i | 1 | -1 | -1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | -1 | -1 | i | -i | -i | -i | -i | i | i | i | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -i | i | 1 | 1 | 1 | -i | i | i | i | i | -i | -i | -i | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | 0 | -2 | 2 | -2 | -2 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -√2 | √2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -2 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | √2 | -√2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | -2 | 0 | 0 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -√-2 | -√-2 | -√-2 | √-2 | √-2 | √-2 | √-2 | -√-2 | -2 | 0 | 0 | complex lifted from SD16 |
| ρ14 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | √-2 | √-2 | √-2 | -√-2 | -√-2 | -√-2 | -√-2 | √-2 | -2 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | -√2 | √2 | √-2 | -√-2 | 0 | 0 | -2 | 0 | 0 | ζ1613+ζ167 | ζ167+ζ165 | ζ1615+ζ1613 | ζ169+ζ163 | ζ1611+ζ16 | ζ163+ζ16 | ζ1611+ζ169 | ζ1615+ζ165 | 0 | √2 | -√2 | complex lifted from D8.C4 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | -√2 | √2 | -√-2 | √-2 | 0 | 0 | -2 | 0 | 0 | ζ1611+ζ16 | ζ163+ζ16 | ζ1611+ζ169 | ζ1615+ζ165 | ζ1613+ζ167 | ζ167+ζ165 | ζ1615+ζ1613 | ζ169+ζ163 | 0 | √2 | -√2 | complex lifted from D8.C4 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | √2 | -√2 | √-2 | -√-2 | 0 | 0 | -2 | 0 | 0 | ζ1615+ζ1613 | ζ1613+ζ167 | ζ1615+ζ165 | ζ163+ζ16 | ζ1611+ζ169 | ζ1611+ζ16 | ζ169+ζ163 | ζ167+ζ165 | 0 | -√2 | √2 | complex lifted from D8.C4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | -√2 | √2 | √-2 | -√-2 | 0 | 0 | -2 | 0 | 0 | ζ1615+ζ165 | ζ1615+ζ1613 | ζ167+ζ165 | ζ1611+ζ16 | ζ169+ζ163 | ζ1611+ζ169 | ζ163+ζ16 | ζ1613+ζ167 | 0 | √2 | -√2 | complex lifted from D8.C4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | √2 | -√2 | -√-2 | √-2 | 0 | 0 | -2 | 0 | 0 | ζ163+ζ16 | ζ169+ζ163 | ζ1611+ζ16 | ζ1615+ζ1613 | ζ167+ζ165 | ζ1615+ζ165 | ζ1613+ζ167 | ζ1611+ζ169 | 0 | -√2 | √2 | complex lifted from D8.C4 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | -√2 | √2 | -√-2 | √-2 | 0 | 0 | -2 | 0 | 0 | ζ169+ζ163 | ζ1611+ζ169 | ζ163+ζ16 | ζ1613+ζ167 | ζ1615+ζ165 | ζ1615+ζ1613 | ζ167+ζ165 | ζ1611+ζ16 | 0 | √2 | -√2 | complex lifted from D8.C4 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | √2 | -√2 | √-2 | -√-2 | 0 | 0 | -2 | 0 | 0 | ζ167+ζ165 | ζ1615+ζ165 | ζ1613+ζ167 | ζ1611+ζ169 | ζ163+ζ16 | ζ169+ζ163 | ζ1611+ζ16 | ζ1615+ζ1613 | 0 | -√2 | √2 | complex lifted from D8.C4 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | √2 | -√2 | -√-2 | √-2 | 0 | 0 | -2 | 0 | 0 | ζ1611+ζ169 | ζ1611+ζ16 | ζ169+ζ163 | ζ167+ζ165 | ζ1615+ζ1613 | ζ1613+ζ167 | ζ1615+ζ165 | ζ163+ζ16 | 0 | -√2 | √2 | complex lifted from D8.C4 |
| ρ23 | 4 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ24 | 4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | -1 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | √5 | -√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ26 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | -1 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | -√5 | √5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from C22⋊F5 |
| ρ27 | 8 | 8 | 0 | 0 | -8 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D20⋊C4, Schur index 2 |
| ρ28 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -4√2 | 4√2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
| ρ29 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4√2 | -4√2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
►On 160 points
Generators in S160
(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)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)
(1 82)(2 81)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 147)(10 146)(11 145)(12 152)(13 151)(14 150)(15 149)(16 148)(17 132)(18 131)(19 130)(20 129)(21 136)(22 135)(23 134)(24 133)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 128)(40 127)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 80)(48 79)(49 90)(50 89)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 104)(64 103)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(113 157)(114 156)(115 155)(116 154)(117 153)(118 160)(119 159)(120 158)
(1 72 44 61 53)(2 65 45 62 54)(3 66 46 63 55)(4 67 47 64 56)(5 68 48 57 49)(6 69 41 58 50)(7 70 42 59 51)(8 71 43 60 52)(9 24 40 158 28)(10 17 33 159 29)(11 18 34 160 30)(12 19 35 153 31)(13 20 36 154 32)(14 21 37 155 25)(15 22 38 156 26)(16 23 39 157 27)(73 104 92 88 111)(74 97 93 81 112)(75 98 94 82 105)(76 99 95 83 106)(77 100 96 84 107)(78 101 89 85 108)(79 102 90 86 109)(80 103 91 87 110)(113 142 148 134 128)(114 143 149 135 121)(115 144 150 136 122)(116 137 151 129 123)(117 138 152 130 124)(118 139 145 131 125)(119 140 146 132 126)(120 141 147 133 127)
(1 121 3 127 5 125 7 123)(2 128 4 126 6 124 8 122)(9 112 30 91 13 108 26 95)(10 111 31 90 14 107 27 94)(11 110 32 89 15 106 28 93)(12 109 25 96 16 105 29 92)(17 104 153 79 21 100 157 75)(18 103 154 78 22 99 158 74)(19 102 155 77 23 98 159 73)(20 101 156 76 24 97 160 80)(33 88 35 86 37 84 39 82)(34 87 36 85 38 83 40 81)(41 130 60 115 45 134 64 119)(42 129 61 114 46 133 57 118)(43 136 62 113 47 132 58 117)(44 135 63 120 48 131 59 116)(49 145 70 137 53 149 66 141)(50 152 71 144 54 148 67 140)(51 151 72 143 55 147 68 139)(52 150 65 142 56 146 69 138)
G:=sub<Sym(160)| (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,82)(2,81)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,147)(10,146)(11,145)(12,152)(13,151)(14,150)(15,149)(16,148)(17,132)(18,131)(19,130)(20,129)(21,136)(22,135)(23,134)(24,133)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,128)(40,127)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,80)(48,79)(49,90)(50,89)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,104)(64,103)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(113,157)(114,156)(115,155)(116,154)(117,153)(118,160)(119,159)(120,158), (1,72,44,61,53)(2,65,45,62,54)(3,66,46,63,55)(4,67,47,64,56)(5,68,48,57,49)(6,69,41,58,50)(7,70,42,59,51)(8,71,43,60,52)(9,24,40,158,28)(10,17,33,159,29)(11,18,34,160,30)(12,19,35,153,31)(13,20,36,154,32)(14,21,37,155,25)(15,22,38,156,26)(16,23,39,157,27)(73,104,92,88,111)(74,97,93,81,112)(75,98,94,82,105)(76,99,95,83,106)(77,100,96,84,107)(78,101,89,85,108)(79,102,90,86,109)(80,103,91,87,110)(113,142,148,134,128)(114,143,149,135,121)(115,144,150,136,122)(116,137,151,129,123)(117,138,152,130,124)(118,139,145,131,125)(119,140,146,132,126)(120,141,147,133,127), (1,121,3,127,5,125,7,123)(2,128,4,126,6,124,8,122)(9,112,30,91,13,108,26,95)(10,111,31,90,14,107,27,94)(11,110,32,89,15,106,28,93)(12,109,25,96,16,105,29,92)(17,104,153,79,21,100,157,75)(18,103,154,78,22,99,158,74)(19,102,155,77,23,98,159,73)(20,101,156,76,24,97,160,80)(33,88,35,86,37,84,39,82)(34,87,36,85,38,83,40,81)(41,130,60,115,45,134,64,119)(42,129,61,114,46,133,57,118)(43,136,62,113,47,132,58,117)(44,135,63,120,48,131,59,116)(49,145,70,137,53,149,66,141)(50,152,71,144,54,148,67,140)(51,151,72,143,55,147,68,139)(52,150,65,142,56,146,69,138)>;
G:=Group( (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)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160), (1,82)(2,81)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,147)(10,146)(11,145)(12,152)(13,151)(14,150)(15,149)(16,148)(17,132)(18,131)(19,130)(20,129)(21,136)(22,135)(23,134)(24,133)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,128)(40,127)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,80)(48,79)(49,90)(50,89)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,102)(58,101)(59,100)(60,99)(61,98)(62,97)(63,104)(64,103)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(113,157)(114,156)(115,155)(116,154)(117,153)(118,160)(119,159)(120,158), (1,72,44,61,53)(2,65,45,62,54)(3,66,46,63,55)(4,67,47,64,56)(5,68,48,57,49)(6,69,41,58,50)(7,70,42,59,51)(8,71,43,60,52)(9,24,40,158,28)(10,17,33,159,29)(11,18,34,160,30)(12,19,35,153,31)(13,20,36,154,32)(14,21,37,155,25)(15,22,38,156,26)(16,23,39,157,27)(73,104,92,88,111)(74,97,93,81,112)(75,98,94,82,105)(76,99,95,83,106)(77,100,96,84,107)(78,101,89,85,108)(79,102,90,86,109)(80,103,91,87,110)(113,142,148,134,128)(114,143,149,135,121)(115,144,150,136,122)(116,137,151,129,123)(117,138,152,130,124)(118,139,145,131,125)(119,140,146,132,126)(120,141,147,133,127), (1,121,3,127,5,125,7,123)(2,128,4,126,6,124,8,122)(9,112,30,91,13,108,26,95)(10,111,31,90,14,107,27,94)(11,110,32,89,15,106,28,93)(12,109,25,96,16,105,29,92)(17,104,153,79,21,100,157,75)(18,103,154,78,22,99,158,74)(19,102,155,77,23,98,159,73)(20,101,156,76,24,97,160,80)(33,88,35,86,37,84,39,82)(34,87,36,85,38,83,40,81)(41,130,60,115,45,134,64,119)(42,129,61,114,46,133,57,118)(43,136,62,113,47,132,58,117)(44,135,63,120,48,131,59,116)(49,145,70,137,53,149,66,141)(50,152,71,144,54,148,67,140)(51,151,72,143,55,147,68,139)(52,150,65,142,56,146,69,138) );
G=PermutationGroup([(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),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)], [(1,82),(2,81),(3,88),(4,87),(5,86),(6,85),(7,84),(8,83),(9,147),(10,146),(11,145),(12,152),(13,151),(14,150),(15,149),(16,148),(17,132),(18,131),(19,130),(20,129),(21,136),(22,135),(23,134),(24,133),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,128),(40,127),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,80),(48,79),(49,90),(50,89),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,102),(58,101),(59,100),(60,99),(61,98),(62,97),(63,104),(64,103),(65,112),(66,111),(67,110),(68,109),(69,108),(70,107),(71,106),(72,105),(113,157),(114,156),(115,155),(116,154),(117,153),(118,160),(119,159),(120,158)], [(1,72,44,61,53),(2,65,45,62,54),(3,66,46,63,55),(4,67,47,64,56),(5,68,48,57,49),(6,69,41,58,50),(7,70,42,59,51),(8,71,43,60,52),(9,24,40,158,28),(10,17,33,159,29),(11,18,34,160,30),(12,19,35,153,31),(13,20,36,154,32),(14,21,37,155,25),(15,22,38,156,26),(16,23,39,157,27),(73,104,92,88,111),(74,97,93,81,112),(75,98,94,82,105),(76,99,95,83,106),(77,100,96,84,107),(78,101,89,85,108),(79,102,90,86,109),(80,103,91,87,110),(113,142,148,134,128),(114,143,149,135,121),(115,144,150,136,122),(116,137,151,129,123),(117,138,152,130,124),(118,139,145,131,125),(119,140,146,132,126),(120,141,147,133,127)], [(1,121,3,127,5,125,7,123),(2,128,4,126,6,124,8,122),(9,112,30,91,13,108,26,95),(10,111,31,90,14,107,27,94),(11,110,32,89,15,106,28,93),(12,109,25,96,16,105,29,92),(17,104,153,79,21,100,157,75),(18,103,154,78,22,99,158,74),(19,102,155,77,23,98,159,73),(20,101,156,76,24,97,160,80),(33,88,35,86,37,84,39,82),(34,87,36,85,38,83,40,81),(41,130,60,115,45,134,64,119),(42,129,61,114,46,133,57,118),(43,136,62,113,47,132,58,117),(44,135,63,120,48,131,59,116),(49,145,70,137,53,149,66,141),(50,152,71,144,54,148,67,140),(51,151,72,143,55,147,68,139),(52,150,65,142,56,146,69,138)])
Matrix representation of D8.F5 ►in GL6(𝔽241)
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 117 | 0 | 234 | 234 |
| 0 | 0 | 7 | 124 | 7 | 0 |
| 0 | 0 | 0 | 7 | 124 | 7 |
| 0 | 0 | 234 | 234 | 0 | 117 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 165 | 0 | 0 | 0 | 0 |
| 115 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 136 | 136 |
| 0 | 0 | 136 | 136 | 0 | 8 |
| 0 | 0 | 105 | 113 | 105 | 0 |
| 0 | 0 | 233 | 128 | 128 | 233 |
G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,117,7,0,234,0,0,0,124,7,234,0,0,234,7,124,0,0,0,234,0,7,117],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[0,115,0,0,0,0,165,0,0,0,0,0,0,0,8,136,105,233,0,0,0,136,113,128,0,0,136,0,105,128,0,0,136,8,0,233] >;
D8.F5 in GAP, Magma, Sage, TeX
D_8.F_5
% in TeX
G:=Group("D8.F5"); // GroupNames label
G:=SmallGroup(320,243);
// by ID
G=gap.SmallGroup(320,243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,184,675,346,192,1684,851,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^5=1,d^4=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of D8.F5 in TeX
Character table of D8.F5 in TeX