metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C10.5C4≀C2, (C2×D4).1F5, (D4×C10).1C4, C2.7(D4⋊F5), (C2×Dic10).5C4, C2.4(C23.F5), C10.C42⋊1C2, (C2×Dic5).106D4, C20.17D4.2C2, C10.3(C4.D4), (C4×Dic5).3C22, C5⋊1(C42.C22), C22.60(C22⋊F5), (C2×C4).15(C2×F5), (C2×C20).10(C2×C4), (C2×C10).34(C22⋊C4), SmallGroup(320,259)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D4).F5
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=a, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >
Subgroups: 298 in 70 conjugacy classes, 22 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×4], C22, C22 [×3], C5, C8 [×4], C2×C4, C2×C4 [×3], D4, Q8, C23, C10, C10 [×2], C10, C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, Dic5 [×3], C20, C2×C10, C2×C10 [×3], C8⋊C4 [×2], C4.4D4, C5⋊C8 [×4], Dic10, C2×Dic5 [×2], C2×Dic5, C2×C20, C5×D4, C22×C10, C42.C22, C4×Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×Dic10, D4×C10, C10.C42 [×2], C20.17D4, (C2×D4).F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C4.D4, C4≀C2 [×2], C2×F5, C42.C22, C22⋊F5, D4⋊F5 [×2], C23.F5, (C2×D4).F5
Character table of (C2×D4).F5
class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 20A | 20B | |
size | 1 | 1 | 1 | 1 | 8 | 4 | 10 | 10 | 10 | 10 | 40 | 4 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 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 | 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 | -i | i | i | -i | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ6 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | -i | i | i | i | i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | i | -i | -i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | i | -i | -i | i | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | -1-i | 0 | 0 | 0 | 0 | -1+i | 1-i | 1+i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 2 | 1+i | 0 | 0 | 0 | 0 | 1-i | -1+i | -1-i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 0 | 1+i | -1-i | 1-i | -1+i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 0 | -1+i | 1-i | -1-i | 1+i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 0 | -1-i | 1+i | -1+i | 1-i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | -1+i | 0 | 0 | 0 | 0 | -1-i | 1+i | 1-i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 2 | 1-i | 0 | 0 | 0 | 0 | 1+i | -1-i | -1+i | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 0 | 1-i | -1+i | 1+i | -1-i | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4≀C2 |
ρ19 | 4 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
ρ20 | 4 | 4 | 4 | 4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×F5 |
ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.D4 |
ρ22 | 4 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | √5 | √5 | -√5 | -√5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
ρ23 | 4 | 4 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -√5 | -√5 | √5 | √5 | 1 | 1 | orthogonal lifted from C22⋊F5 |
ρ24 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | -√5 | √5 | complex lifted from C23.F5 |
ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | -√5 | √5 | complex lifted from C23.F5 |
ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ53+2ζ5+1 | 2ζ54+2ζ52+1 | 2ζ54+2ζ53+1 | 2ζ52+2ζ5+1 | √5 | -√5 | complex lifted from C23.F5 |
ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 2ζ54+2ζ52+1 | 2ζ53+2ζ5+1 | 2ζ52+2ζ5+1 | 2ζ54+2ζ53+1 | √5 | -√5 | complex lifted from C23.F5 |
ρ28 | 8 | 8 | -8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊F5, Schur index 2 |
ρ29 | 8 | -8 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊F5, Schur index 2 |
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)
(1 128 88 115)(2 125 81 120)(3 122 82 117)(4 127 83 114)(5 124 84 119)(6 121 85 116)(7 126 86 113)(8 123 87 118)(9 97 106 30)(10 102 107 27)(11 99 108 32)(12 104 109 29)(13 101 110 26)(14 98 111 31)(15 103 112 28)(16 100 105 25)(17 131 152 153)(18 136 145 158)(19 133 146 155)(20 130 147 160)(21 135 148 157)(22 132 149 154)(23 129 150 159)(24 134 151 156)(33 51 68 91)(34 56 69 96)(35 53 70 93)(36 50 71 90)(37 55 72 95)(38 52 65 92)(39 49 66 89)(40 54 67 94)(41 144 76 62)(42 141 77 59)(43 138 78 64)(44 143 79 61)(45 140 80 58)(46 137 73 63)(47 142 74 60)(48 139 75 57)
(2 121)(3 86)(4 114)(6 125)(7 82)(8 118)(9 110)(10 27)(12 100)(13 106)(14 31)(16 104)(18 132)(19 150)(20 160)(22 136)(23 146)(24 156)(25 109)(26 30)(28 103)(29 105)(32 99)(34 52)(35 66)(36 90)(38 56)(39 70)(40 94)(41 58)(42 46)(43 138)(44 79)(45 62)(47 142)(48 75)(49 53)(50 71)(51 91)(54 67)(55 95)(59 137)(60 74)(63 141)(64 78)(65 96)(69 92)(73 77)(76 140)(80 144)(81 116)(83 127)(85 120)(87 123)(89 93)(97 101)(98 111)(102 107)(113 117)(115 128)(119 124)(122 126)(129 133)(130 147)(131 153)(134 151)(135 157)(145 154)(149 158)(155 159)
(1 143 108 37 17)(2 38 144 18 109)(3 19 39 110 137)(4 111 20 138 40)(5 139 112 33 21)(6 34 140 22 105)(7 23 35 106 141)(8 107 24 142 36)(9 59 86 150 70)(10 151 60 71 87)(11 72 152 88 61)(12 81 65 62 145)(13 63 82 146 66)(14 147 64 67 83)(15 68 148 84 57)(16 85 69 58 149)(25 121 56 80 132)(26 73 122 133 49)(27 134 74 50 123)(28 51 135 124 75)(29 125 52 76 136)(30 77 126 129 53)(31 130 78 54 127)(32 55 131 128 79)(41 158 104 120 92)(42 113 159 93 97)(43 94 114 98 160)(44 99 95 153 115)(45 154 100 116 96)(46 117 155 89 101)(47 90 118 102 156)(48 103 91 157 119)
(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)
G:=sub<Sym(160)| (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,128,88,115)(2,125,81,120)(3,122,82,117)(4,127,83,114)(5,124,84,119)(6,121,85,116)(7,126,86,113)(8,123,87,118)(9,97,106,30)(10,102,107,27)(11,99,108,32)(12,104,109,29)(13,101,110,26)(14,98,111,31)(15,103,112,28)(16,100,105,25)(17,131,152,153)(18,136,145,158)(19,133,146,155)(20,130,147,160)(21,135,148,157)(22,132,149,154)(23,129,150,159)(24,134,151,156)(33,51,68,91)(34,56,69,96)(35,53,70,93)(36,50,71,90)(37,55,72,95)(38,52,65,92)(39,49,66,89)(40,54,67,94)(41,144,76,62)(42,141,77,59)(43,138,78,64)(44,143,79,61)(45,140,80,58)(46,137,73,63)(47,142,74,60)(48,139,75,57), (2,121)(3,86)(4,114)(6,125)(7,82)(8,118)(9,110)(10,27)(12,100)(13,106)(14,31)(16,104)(18,132)(19,150)(20,160)(22,136)(23,146)(24,156)(25,109)(26,30)(28,103)(29,105)(32,99)(34,52)(35,66)(36,90)(38,56)(39,70)(40,94)(41,58)(42,46)(43,138)(44,79)(45,62)(47,142)(48,75)(49,53)(50,71)(51,91)(54,67)(55,95)(59,137)(60,74)(63,141)(64,78)(65,96)(69,92)(73,77)(76,140)(80,144)(81,116)(83,127)(85,120)(87,123)(89,93)(97,101)(98,111)(102,107)(113,117)(115,128)(119,124)(122,126)(129,133)(130,147)(131,153)(134,151)(135,157)(145,154)(149,158)(155,159), (1,143,108,37,17)(2,38,144,18,109)(3,19,39,110,137)(4,111,20,138,40)(5,139,112,33,21)(6,34,140,22,105)(7,23,35,106,141)(8,107,24,142,36)(9,59,86,150,70)(10,151,60,71,87)(11,72,152,88,61)(12,81,65,62,145)(13,63,82,146,66)(14,147,64,67,83)(15,68,148,84,57)(16,85,69,58,149)(25,121,56,80,132)(26,73,122,133,49)(27,134,74,50,123)(28,51,135,124,75)(29,125,52,76,136)(30,77,126,129,53)(31,130,78,54,127)(32,55,131,128,79)(41,158,104,120,92)(42,113,159,93,97)(43,94,114,98,160)(44,99,95,153,115)(45,154,100,116,96)(46,117,155,89,101)(47,90,118,102,156)(48,103,91,157,119), (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)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160), (1,128,88,115)(2,125,81,120)(3,122,82,117)(4,127,83,114)(5,124,84,119)(6,121,85,116)(7,126,86,113)(8,123,87,118)(9,97,106,30)(10,102,107,27)(11,99,108,32)(12,104,109,29)(13,101,110,26)(14,98,111,31)(15,103,112,28)(16,100,105,25)(17,131,152,153)(18,136,145,158)(19,133,146,155)(20,130,147,160)(21,135,148,157)(22,132,149,154)(23,129,150,159)(24,134,151,156)(33,51,68,91)(34,56,69,96)(35,53,70,93)(36,50,71,90)(37,55,72,95)(38,52,65,92)(39,49,66,89)(40,54,67,94)(41,144,76,62)(42,141,77,59)(43,138,78,64)(44,143,79,61)(45,140,80,58)(46,137,73,63)(47,142,74,60)(48,139,75,57), (2,121)(3,86)(4,114)(6,125)(7,82)(8,118)(9,110)(10,27)(12,100)(13,106)(14,31)(16,104)(18,132)(19,150)(20,160)(22,136)(23,146)(24,156)(25,109)(26,30)(28,103)(29,105)(32,99)(34,52)(35,66)(36,90)(38,56)(39,70)(40,94)(41,58)(42,46)(43,138)(44,79)(45,62)(47,142)(48,75)(49,53)(50,71)(51,91)(54,67)(55,95)(59,137)(60,74)(63,141)(64,78)(65,96)(69,92)(73,77)(76,140)(80,144)(81,116)(83,127)(85,120)(87,123)(89,93)(97,101)(98,111)(102,107)(113,117)(115,128)(119,124)(122,126)(129,133)(130,147)(131,153)(134,151)(135,157)(145,154)(149,158)(155,159), (1,143,108,37,17)(2,38,144,18,109)(3,19,39,110,137)(4,111,20,138,40)(5,139,112,33,21)(6,34,140,22,105)(7,23,35,106,141)(8,107,24,142,36)(9,59,86,150,70)(10,151,60,71,87)(11,72,152,88,61)(12,81,65,62,145)(13,63,82,146,66)(14,147,64,67,83)(15,68,148,84,57)(16,85,69,58,149)(25,121,56,80,132)(26,73,122,133,49)(27,134,74,50,123)(28,51,135,124,75)(29,125,52,76,136)(30,77,126,129,53)(31,130,78,54,127)(32,55,131,128,79)(41,158,104,120,92)(42,113,159,93,97)(43,94,114,98,160)(44,99,95,153,115)(45,154,100,116,96)(46,117,155,89,101)(47,90,118,102,156)(48,103,91,157,119), (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) );
G=PermutationGroup([(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160)], [(1,128,88,115),(2,125,81,120),(3,122,82,117),(4,127,83,114),(5,124,84,119),(6,121,85,116),(7,126,86,113),(8,123,87,118),(9,97,106,30),(10,102,107,27),(11,99,108,32),(12,104,109,29),(13,101,110,26),(14,98,111,31),(15,103,112,28),(16,100,105,25),(17,131,152,153),(18,136,145,158),(19,133,146,155),(20,130,147,160),(21,135,148,157),(22,132,149,154),(23,129,150,159),(24,134,151,156),(33,51,68,91),(34,56,69,96),(35,53,70,93),(36,50,71,90),(37,55,72,95),(38,52,65,92),(39,49,66,89),(40,54,67,94),(41,144,76,62),(42,141,77,59),(43,138,78,64),(44,143,79,61),(45,140,80,58),(46,137,73,63),(47,142,74,60),(48,139,75,57)], [(2,121),(3,86),(4,114),(6,125),(7,82),(8,118),(9,110),(10,27),(12,100),(13,106),(14,31),(16,104),(18,132),(19,150),(20,160),(22,136),(23,146),(24,156),(25,109),(26,30),(28,103),(29,105),(32,99),(34,52),(35,66),(36,90),(38,56),(39,70),(40,94),(41,58),(42,46),(43,138),(44,79),(45,62),(47,142),(48,75),(49,53),(50,71),(51,91),(54,67),(55,95),(59,137),(60,74),(63,141),(64,78),(65,96),(69,92),(73,77),(76,140),(80,144),(81,116),(83,127),(85,120),(87,123),(89,93),(97,101),(98,111),(102,107),(113,117),(115,128),(119,124),(122,126),(129,133),(130,147),(131,153),(134,151),(135,157),(145,154),(149,158),(155,159)], [(1,143,108,37,17),(2,38,144,18,109),(3,19,39,110,137),(4,111,20,138,40),(5,139,112,33,21),(6,34,140,22,105),(7,23,35,106,141),(8,107,24,142,36),(9,59,86,150,70),(10,151,60,71,87),(11,72,152,88,61),(12,81,65,62,145),(13,63,82,146,66),(14,147,64,67,83),(15,68,148,84,57),(16,85,69,58,149),(25,121,56,80,132),(26,73,122,133,49),(27,134,74,50,123),(28,51,135,124,75),(29,125,52,76,136),(30,77,126,129,53),(31,130,78,54,127),(32,55,131,128,79),(41,158,104,120,92),(42,113,159,93,97),(43,94,114,98,160),(44,99,95,153,115),(45,154,100,116,96),(46,117,155,89,101),(47,90,118,102,156),(48,103,91,157,119)], [(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)])
Matrix representation of (C2×D4).F5 ►in GL8(𝔽41)
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 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 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 34 | 30 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 7 | 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 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 36 | 40 | 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 |
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 | 40 | 40 | 40 | 40 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
36 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
5 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 14 | 0 | 0 | 0 | 0 |
0 | 0 | 24 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 16 | 3 | 34 | 14 |
0 | 0 | 0 | 0 | 31 | 11 | 38 | 13 |
0 | 0 | 0 | 0 | 27 | 2 | 30 | 20 |
0 | 0 | 0 | 0 | 28 | 18 | 39 | 25 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,34,12,0,0,0,0,0,0,30,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,36,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[36,5,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,40,24,0,0,0,0,0,0,14,1,0,0,0,0,0,0,0,0,16,31,27,28,0,0,0,0,3,11,2,18,0,0,0,0,34,38,30,39,0,0,0,0,14,13,20,25] >;
(C2×D4).F5 in GAP, Magma, Sage, TeX
(C_2\times D_4).F_5
% in TeX
G:=Group("(C2xD4).F5");
// GroupNames label
G:=SmallGroup(320,259);
// by ID
G=gap.SmallGroup(320,259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,268,1571,570,136,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=a,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^3>;
// generators/relations
Export