Copied to
clipboard

## G = (C2×D4).F5order 320 = 26·5

### 1st non-split extension by C2×D4 of F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×D4).F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C10.C42 — (C2×D4).F5
 Lower central C5 — C2×C10 — C2×C20 — (C2×D4).F5
 Upper central C1 — C22 — C2×C4 — C2×D4

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, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C8⋊C4, C4.4D4, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C22×C10, C42.C22, C4×Dic5, C23.D5, C2×C5⋊C8, C2×Dic10, D4×C10, C10.C42, C20.17D4, (C2×D4).F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C4.D4, C4≀C2, C2×F5, C42.C22, C22⋊F5, D4⋊F5, 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

Smallest permutation representation of (C2×D4).F5
On 160 points
Generators in S160
(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 63 143 35)(2 60 144 40)(3 57 137 37)(4 62 138 34)(5 59 139 39)(6 64 140 36)(7 61 141 33)(8 58 142 38)(9 53 21 82)(10 50 22 87)(11 55 23 84)(12 52 24 81)(13 49 17 86)(14 54 18 83)(15 51 19 88)(16 56 20 85)(25 96 156 108)(26 93 157 105)(27 90 158 110)(28 95 159 107)(29 92 160 112)(30 89 153 109)(31 94 154 106)(32 91 155 111)(41 74 132 69)(42 79 133 66)(43 76 134 71)(44 73 135 68)(45 78 136 65)(46 75 129 70)(47 80 130 67)(48 77 131 72)(97 114 121 148)(98 119 122 145)(99 116 123 150)(100 113 124 147)(101 118 125 152)(102 115 126 149)(103 120 127 146)(104 117 128 151)
(2 64)(3 141)(4 34)(6 60)(7 137)(8 38)(9 17)(10 87)(12 56)(13 21)(14 83)(16 52)(18 54)(20 81)(22 50)(24 85)(25 92)(26 153)(27 110)(29 96)(30 157)(31 106)(33 37)(35 63)(36 144)(39 59)(40 140)(41 65)(42 46)(43 76)(44 135)(45 69)(47 80)(48 131)(49 53)(51 88)(55 84)(57 61)(58 142)(62 138)(66 75)(67 130)(70 79)(71 134)(74 136)(78 132)(82 86)(89 93)(90 158)(91 111)(94 154)(95 107)(97 118)(98 126)(99 150)(101 114)(102 122)(103 146)(105 109)(108 160)(112 156)(113 147)(115 119)(116 123)(117 151)(120 127)(121 152)(125 148)(129 133)(145 149)
(1 19 100 28 77)(2 29 20 78 101)(3 79 30 102 21)(4 103 80 22 31)(5 23 104 32 73)(6 25 24 74 97)(7 75 26 98 17)(8 99 76 18 27)(9 137 66 153 126)(10 154 138 127 67)(11 128 155 68 139)(12 69 121 140 156)(13 141 70 157 122)(14 158 142 123 71)(15 124 159 72 143)(16 65 125 144 160)(33 46 105 145 49)(34 146 47 50 106)(35 51 147 107 48)(36 108 52 41 148)(37 42 109 149 53)(38 150 43 54 110)(39 55 151 111 44)(40 112 56 45 152)(57 133 89 115 82)(58 116 134 83 90)(59 84 117 91 135)(60 92 85 136 118)(61 129 93 119 86)(62 120 130 87 94)(63 88 113 95 131)(64 96 81 132 114)
(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), (1,63,143,35)(2,60,144,40)(3,57,137,37)(4,62,138,34)(5,59,139,39)(6,64,140,36)(7,61,141,33)(8,58,142,38)(9,53,21,82)(10,50,22,87)(11,55,23,84)(12,52,24,81)(13,49,17,86)(14,54,18,83)(15,51,19,88)(16,56,20,85)(25,96,156,108)(26,93,157,105)(27,90,158,110)(28,95,159,107)(29,92,160,112)(30,89,153,109)(31,94,154,106)(32,91,155,111)(41,74,132,69)(42,79,133,66)(43,76,134,71)(44,73,135,68)(45,78,136,65)(46,75,129,70)(47,80,130,67)(48,77,131,72)(97,114,121,148)(98,119,122,145)(99,116,123,150)(100,113,124,147)(101,118,125,152)(102,115,126,149)(103,120,127,146)(104,117,128,151), (2,64)(3,141)(4,34)(6,60)(7,137)(8,38)(9,17)(10,87)(12,56)(13,21)(14,83)(16,52)(18,54)(20,81)(22,50)(24,85)(25,92)(26,153)(27,110)(29,96)(30,157)(31,106)(33,37)(35,63)(36,144)(39,59)(40,140)(41,65)(42,46)(43,76)(44,135)(45,69)(47,80)(48,131)(49,53)(51,88)(55,84)(57,61)(58,142)(62,138)(66,75)(67,130)(70,79)(71,134)(74,136)(78,132)(82,86)(89,93)(90,158)(91,111)(94,154)(95,107)(97,118)(98,126)(99,150)(101,114)(102,122)(103,146)(105,109)(108,160)(112,156)(113,147)(115,119)(116,123)(117,151)(120,127)(121,152)(125,148)(129,133)(145,149), (1,19,100,28,77)(2,29,20,78,101)(3,79,30,102,21)(4,103,80,22,31)(5,23,104,32,73)(6,25,24,74,97)(7,75,26,98,17)(8,99,76,18,27)(9,137,66,153,126)(10,154,138,127,67)(11,128,155,68,139)(12,69,121,140,156)(13,141,70,157,122)(14,158,142,123,71)(15,124,159,72,143)(16,65,125,144,160)(33,46,105,145,49)(34,146,47,50,106)(35,51,147,107,48)(36,108,52,41,148)(37,42,109,149,53)(38,150,43,54,110)(39,55,151,111,44)(40,112,56,45,152)(57,133,89,115,82)(58,116,134,83,90)(59,84,117,91,135)(60,92,85,136,118)(61,129,93,119,86)(62,120,130,87,94)(63,88,113,95,131)(64,96,81,132,114), (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), (1,63,143,35)(2,60,144,40)(3,57,137,37)(4,62,138,34)(5,59,139,39)(6,64,140,36)(7,61,141,33)(8,58,142,38)(9,53,21,82)(10,50,22,87)(11,55,23,84)(12,52,24,81)(13,49,17,86)(14,54,18,83)(15,51,19,88)(16,56,20,85)(25,96,156,108)(26,93,157,105)(27,90,158,110)(28,95,159,107)(29,92,160,112)(30,89,153,109)(31,94,154,106)(32,91,155,111)(41,74,132,69)(42,79,133,66)(43,76,134,71)(44,73,135,68)(45,78,136,65)(46,75,129,70)(47,80,130,67)(48,77,131,72)(97,114,121,148)(98,119,122,145)(99,116,123,150)(100,113,124,147)(101,118,125,152)(102,115,126,149)(103,120,127,146)(104,117,128,151), (2,64)(3,141)(4,34)(6,60)(7,137)(8,38)(9,17)(10,87)(12,56)(13,21)(14,83)(16,52)(18,54)(20,81)(22,50)(24,85)(25,92)(26,153)(27,110)(29,96)(30,157)(31,106)(33,37)(35,63)(36,144)(39,59)(40,140)(41,65)(42,46)(43,76)(44,135)(45,69)(47,80)(48,131)(49,53)(51,88)(55,84)(57,61)(58,142)(62,138)(66,75)(67,130)(70,79)(71,134)(74,136)(78,132)(82,86)(89,93)(90,158)(91,111)(94,154)(95,107)(97,118)(98,126)(99,150)(101,114)(102,122)(103,146)(105,109)(108,160)(112,156)(113,147)(115,119)(116,123)(117,151)(120,127)(121,152)(125,148)(129,133)(145,149), (1,19,100,28,77)(2,29,20,78,101)(3,79,30,102,21)(4,103,80,22,31)(5,23,104,32,73)(6,25,24,74,97)(7,75,26,98,17)(8,99,76,18,27)(9,137,66,153,126)(10,154,138,127,67)(11,128,155,68,139)(12,69,121,140,156)(13,141,70,157,122)(14,158,142,123,71)(15,124,159,72,143)(16,65,125,144,160)(33,46,105,145,49)(34,146,47,50,106)(35,51,147,107,48)(36,108,52,41,148)(37,42,109,149,53)(38,150,43,54,110)(39,55,151,111,44)(40,112,56,45,152)(57,133,89,115,82)(58,116,134,83,90)(59,84,117,91,135)(60,92,85,136,118)(61,129,93,119,86)(62,120,130,87,94)(63,88,113,95,131)(64,96,81,132,114), (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)], [(1,63,143,35),(2,60,144,40),(3,57,137,37),(4,62,138,34),(5,59,139,39),(6,64,140,36),(7,61,141,33),(8,58,142,38),(9,53,21,82),(10,50,22,87),(11,55,23,84),(12,52,24,81),(13,49,17,86),(14,54,18,83),(15,51,19,88),(16,56,20,85),(25,96,156,108),(26,93,157,105),(27,90,158,110),(28,95,159,107),(29,92,160,112),(30,89,153,109),(31,94,154,106),(32,91,155,111),(41,74,132,69),(42,79,133,66),(43,76,134,71),(44,73,135,68),(45,78,136,65),(46,75,129,70),(47,80,130,67),(48,77,131,72),(97,114,121,148),(98,119,122,145),(99,116,123,150),(100,113,124,147),(101,118,125,152),(102,115,126,149),(103,120,127,146),(104,117,128,151)], [(2,64),(3,141),(4,34),(6,60),(7,137),(8,38),(9,17),(10,87),(12,56),(13,21),(14,83),(16,52),(18,54),(20,81),(22,50),(24,85),(25,92),(26,153),(27,110),(29,96),(30,157),(31,106),(33,37),(35,63),(36,144),(39,59),(40,140),(41,65),(42,46),(43,76),(44,135),(45,69),(47,80),(48,131),(49,53),(51,88),(55,84),(57,61),(58,142),(62,138),(66,75),(67,130),(70,79),(71,134),(74,136),(78,132),(82,86),(89,93),(90,158),(91,111),(94,154),(95,107),(97,118),(98,126),(99,150),(101,114),(102,122),(103,146),(105,109),(108,160),(112,156),(113,147),(115,119),(116,123),(117,151),(120,127),(121,152),(125,148),(129,133),(145,149)], [(1,19,100,28,77),(2,29,20,78,101),(3,79,30,102,21),(4,103,80,22,31),(5,23,104,32,73),(6,25,24,74,97),(7,75,26,98,17),(8,99,76,18,27),(9,137,66,153,126),(10,154,138,127,67),(11,128,155,68,139),(12,69,121,140,156),(13,141,70,157,122),(14,158,142,123,71),(15,124,159,72,143),(16,65,125,144,160),(33,46,105,145,49),(34,146,47,50,106),(35,51,147,107,48),(36,108,52,41,148),(37,42,109,149,53),(38,150,43,54,110),(39,55,151,111,44),(40,112,56,45,152),(57,133,89,115,82),(58,116,134,83,90),(59,84,117,91,135),(60,92,85,136,118),(61,129,93,119,86),(62,120,130,87,94),(63,88,113,95,131),(64,96,81,132,114)], [(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

׿
×
𝔽