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## G = (C2×Q8).F5order 320 = 26·5

### 1st non-split extension by C2×Q8 of F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×Q8).F5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C10.C42 — (C2×Q8).F5
 Lower central C5 — C2×C10 — C2×C20 — (C2×Q8).F5
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for (C2×Q8).F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=b2, e4=a, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 362 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, D5, C10, C10 [×2], C42, C22⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×3], C2×C10, C8⋊C4 [×2], C4.4D4, C5⋊C8 [×4], D20, C2×Dic5 [×2], C2×C20, C2×C20, C5×Q8, C22×D5, C42.C22, C4×Dic5, D10⋊C4 [×2], C2×C5⋊C8 [×2], C2×D20, Q8×C10, C10.C42 [×2], C20.23D4, (C2×Q8).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, Q82F5 [×2], C23.F5, (C2×Q8).F5

Character table of (C2×Q8).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 20A 20B 20C 20D 20E 20F size 1 1 1 1 40 4 8 10 10 10 10 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 0 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 2 2 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -2 0 -2 2 2 -2 2 0 0 0 0 0 0 0 0 2 2 2 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 0 2i 0 0 -2i 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 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 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 ρ14 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 ρ15 2 -2 -2 2 0 0 0 -2i 0 0 2i 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 2i 0 0 -2i 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 ρ18 2 -2 -2 2 0 0 0 -2i 0 0 2i 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 0 4 4 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 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 ρ21 4 4 4 4 0 4 -4 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 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 1 1 √5 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 1 1 -√5 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+2ζ52+1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 -√5 √5 2ζ52+2ζ5+1 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 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 -√5 √5 2ζ54+2ζ53+1 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 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 √5 -√5 2ζ53+2ζ5+1 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+2ζ53+1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 √5 -√5 2ζ54+2ζ52+1 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 orthogonal lifted from Q8⋊2F5 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 orthogonal lifted from Q8⋊2F5

Smallest permutation representation of (C2×Q8).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 88 110 115)(2 85 111 120)(3 82 112 117)(4 87 105 114)(5 84 106 119)(6 81 107 116)(7 86 108 113)(8 83 109 118)(9 104 152 26)(10 101 145 31)(11 98 146 28)(12 103 147 25)(13 100 148 30)(14 97 149 27)(15 102 150 32)(16 99 151 29)(17 134 38 78)(18 131 39 75)(19 136 40 80)(20 133 33 77)(21 130 34 74)(22 135 35 79)(23 132 36 76)(24 129 37 73)(41 60 56 125)(42 57 49 122)(43 62 50 127)(44 59 51 124)(45 64 52 121)(46 61 53 126)(47 58 54 123)(48 63 55 128)(65 143 155 93)(66 140 156 90)(67 137 157 95)(68 142 158 92)(69 139 159 89)(70 144 160 94)(71 141 153 91)(72 138 154 96)
(1 112 110 3)(2 118 111 83)(4 81 105 116)(5 108 106 7)(6 114 107 87)(8 85 109 120)(9 135 152 79)(10 36 145 23)(11 77 146 133)(12 21 147 34)(13 131 148 75)(14 40 149 19)(15 73 150 129)(16 17 151 38)(18 30 39 100)(20 98 33 28)(22 26 35 104)(24 102 37 32)(25 130 103 74)(27 80 97 136)(29 134 99 78)(31 76 101 132)(41 144 56 94)(42 153 49 71)(43 92 50 142)(44 69 51 159)(45 140 52 90)(46 157 53 67)(47 96 54 138)(48 65 55 155)(57 141 122 91)(58 154 123 72)(59 89 124 139)(60 70 125 160)(61 137 126 95)(62 158 127 68)(63 93 128 143)(64 66 121 156)(82 115 117 88)(84 86 119 113)
(1 98 141 63 18)(2 64 99 19 142)(3 20 57 143 100)(4 144 21 101 58)(5 102 137 59 22)(6 60 103 23 138)(7 24 61 139 104)(8 140 17 97 62)(9 113 73 46 69)(10 47 114 70 74)(11 71 48 75 115)(12 76 72 116 41)(13 117 77 42 65)(14 43 118 66 78)(15 67 44 79 119)(16 80 68 120 45)(25 36 96 107 125)(26 108 37 126 89)(27 127 109 90 38)(28 91 128 39 110)(29 40 92 111 121)(30 112 33 122 93)(31 123 105 94 34)(32 95 124 35 106)(49 155 148 82 133)(50 83 156 134 149)(51 135 84 150 157)(52 151 136 158 85)(53 159 152 86 129)(54 87 160 130 145)(55 131 88 146 153)(56 147 132 154 81)
(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), (1,88,110,115)(2,85,111,120)(3,82,112,117)(4,87,105,114)(5,84,106,119)(6,81,107,116)(7,86,108,113)(8,83,109,118)(9,104,152,26)(10,101,145,31)(11,98,146,28)(12,103,147,25)(13,100,148,30)(14,97,149,27)(15,102,150,32)(16,99,151,29)(17,134,38,78)(18,131,39,75)(19,136,40,80)(20,133,33,77)(21,130,34,74)(22,135,35,79)(23,132,36,76)(24,129,37,73)(41,60,56,125)(42,57,49,122)(43,62,50,127)(44,59,51,124)(45,64,52,121)(46,61,53,126)(47,58,54,123)(48,63,55,128)(65,143,155,93)(66,140,156,90)(67,137,157,95)(68,142,158,92)(69,139,159,89)(70,144,160,94)(71,141,153,91)(72,138,154,96), (1,112,110,3)(2,118,111,83)(4,81,105,116)(5,108,106,7)(6,114,107,87)(8,85,109,120)(9,135,152,79)(10,36,145,23)(11,77,146,133)(12,21,147,34)(13,131,148,75)(14,40,149,19)(15,73,150,129)(16,17,151,38)(18,30,39,100)(20,98,33,28)(22,26,35,104)(24,102,37,32)(25,130,103,74)(27,80,97,136)(29,134,99,78)(31,76,101,132)(41,144,56,94)(42,153,49,71)(43,92,50,142)(44,69,51,159)(45,140,52,90)(46,157,53,67)(47,96,54,138)(48,65,55,155)(57,141,122,91)(58,154,123,72)(59,89,124,139)(60,70,125,160)(61,137,126,95)(62,158,127,68)(63,93,128,143)(64,66,121,156)(82,115,117,88)(84,86,119,113), (1,98,141,63,18)(2,64,99,19,142)(3,20,57,143,100)(4,144,21,101,58)(5,102,137,59,22)(6,60,103,23,138)(7,24,61,139,104)(8,140,17,97,62)(9,113,73,46,69)(10,47,114,70,74)(11,71,48,75,115)(12,76,72,116,41)(13,117,77,42,65)(14,43,118,66,78)(15,67,44,79,119)(16,80,68,120,45)(25,36,96,107,125)(26,108,37,126,89)(27,127,109,90,38)(28,91,128,39,110)(29,40,92,111,121)(30,112,33,122,93)(31,123,105,94,34)(32,95,124,35,106)(49,155,148,82,133)(50,83,156,134,149)(51,135,84,150,157)(52,151,136,158,85)(53,159,152,86,129)(54,87,160,130,145)(55,131,88,146,153)(56,147,132,154,81), (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), (1,88,110,115)(2,85,111,120)(3,82,112,117)(4,87,105,114)(5,84,106,119)(6,81,107,116)(7,86,108,113)(8,83,109,118)(9,104,152,26)(10,101,145,31)(11,98,146,28)(12,103,147,25)(13,100,148,30)(14,97,149,27)(15,102,150,32)(16,99,151,29)(17,134,38,78)(18,131,39,75)(19,136,40,80)(20,133,33,77)(21,130,34,74)(22,135,35,79)(23,132,36,76)(24,129,37,73)(41,60,56,125)(42,57,49,122)(43,62,50,127)(44,59,51,124)(45,64,52,121)(46,61,53,126)(47,58,54,123)(48,63,55,128)(65,143,155,93)(66,140,156,90)(67,137,157,95)(68,142,158,92)(69,139,159,89)(70,144,160,94)(71,141,153,91)(72,138,154,96), (1,112,110,3)(2,118,111,83)(4,81,105,116)(5,108,106,7)(6,114,107,87)(8,85,109,120)(9,135,152,79)(10,36,145,23)(11,77,146,133)(12,21,147,34)(13,131,148,75)(14,40,149,19)(15,73,150,129)(16,17,151,38)(18,30,39,100)(20,98,33,28)(22,26,35,104)(24,102,37,32)(25,130,103,74)(27,80,97,136)(29,134,99,78)(31,76,101,132)(41,144,56,94)(42,153,49,71)(43,92,50,142)(44,69,51,159)(45,140,52,90)(46,157,53,67)(47,96,54,138)(48,65,55,155)(57,141,122,91)(58,154,123,72)(59,89,124,139)(60,70,125,160)(61,137,126,95)(62,158,127,68)(63,93,128,143)(64,66,121,156)(82,115,117,88)(84,86,119,113), (1,98,141,63,18)(2,64,99,19,142)(3,20,57,143,100)(4,144,21,101,58)(5,102,137,59,22)(6,60,103,23,138)(7,24,61,139,104)(8,140,17,97,62)(9,113,73,46,69)(10,47,114,70,74)(11,71,48,75,115)(12,76,72,116,41)(13,117,77,42,65)(14,43,118,66,78)(15,67,44,79,119)(16,80,68,120,45)(25,36,96,107,125)(26,108,37,126,89)(27,127,109,90,38)(28,91,128,39,110)(29,40,92,111,121)(30,112,33,122,93)(31,123,105,94,34)(32,95,124,35,106)(49,155,148,82,133)(50,83,156,134,149)(51,135,84,150,157)(52,151,136,158,85)(53,159,152,86,129)(54,87,160,130,145)(55,131,88,146,153)(56,147,132,154,81), (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)], [(1,88,110,115),(2,85,111,120),(3,82,112,117),(4,87,105,114),(5,84,106,119),(6,81,107,116),(7,86,108,113),(8,83,109,118),(9,104,152,26),(10,101,145,31),(11,98,146,28),(12,103,147,25),(13,100,148,30),(14,97,149,27),(15,102,150,32),(16,99,151,29),(17,134,38,78),(18,131,39,75),(19,136,40,80),(20,133,33,77),(21,130,34,74),(22,135,35,79),(23,132,36,76),(24,129,37,73),(41,60,56,125),(42,57,49,122),(43,62,50,127),(44,59,51,124),(45,64,52,121),(46,61,53,126),(47,58,54,123),(48,63,55,128),(65,143,155,93),(66,140,156,90),(67,137,157,95),(68,142,158,92),(69,139,159,89),(70,144,160,94),(71,141,153,91),(72,138,154,96)], [(1,112,110,3),(2,118,111,83),(4,81,105,116),(5,108,106,7),(6,114,107,87),(8,85,109,120),(9,135,152,79),(10,36,145,23),(11,77,146,133),(12,21,147,34),(13,131,148,75),(14,40,149,19),(15,73,150,129),(16,17,151,38),(18,30,39,100),(20,98,33,28),(22,26,35,104),(24,102,37,32),(25,130,103,74),(27,80,97,136),(29,134,99,78),(31,76,101,132),(41,144,56,94),(42,153,49,71),(43,92,50,142),(44,69,51,159),(45,140,52,90),(46,157,53,67),(47,96,54,138),(48,65,55,155),(57,141,122,91),(58,154,123,72),(59,89,124,139),(60,70,125,160),(61,137,126,95),(62,158,127,68),(63,93,128,143),(64,66,121,156),(82,115,117,88),(84,86,119,113)], [(1,98,141,63,18),(2,64,99,19,142),(3,20,57,143,100),(4,144,21,101,58),(5,102,137,59,22),(6,60,103,23,138),(7,24,61,139,104),(8,140,17,97,62),(9,113,73,46,69),(10,47,114,70,74),(11,71,48,75,115),(12,76,72,116,41),(13,117,77,42,65),(14,43,118,66,78),(15,67,44,79,119),(16,80,68,120,45),(25,36,96,107,125),(26,108,37,126,89),(27,127,109,90,38),(28,91,128,39,110),(29,40,92,111,121),(30,112,33,122,93),(31,123,105,94,34),(32,95,124,35,106),(49,155,148,82,133),(50,83,156,134,149),(51,135,84,150,157),(52,151,136,158,85),(53,159,152,86,129),(54,87,160,130,145),(55,131,88,146,153),(56,147,132,154,81)], [(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×Q8).F5 in GL8(𝔽41)

 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 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 0 40 0 0 0 0 0 0 1 0 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 40 0 0 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 19 3 0 38 0 0 0 0 0 22 3 38 0 0 0 0 38 3 22 0 0 0 0 0 38 0 3 19
,
 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 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 36 5 0 0 0 0 0 0 5 5 0 0 0 0 0 0 0 0 37 37 0 0 0 0 0 0 4 37 0 0 0 0 0 0 0 0 34 34 4 6 0 0 0 0 38 40 8 40 0 0 0 0 1 33 1 3 0 0 0 0 35 37 7 7

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[36,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,37,4,0,0,0,0,0,0,37,37,0,0,0,0,0,0,0,0,34,38,1,35,0,0,0,0,34,40,33,37,0,0,0,0,4,8,1,7,0,0,0,0,6,40,3,7] >;

(C2×Q8).F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8).F_5
% in TeX

G:=Group("(C2xQ8).F5");
// GroupNames label

G:=SmallGroup(320,265);
// by ID

G=gap.SmallGroup(320,265);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,268,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=b^2,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^-1=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

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