Q16.F5 - GroupNames

class	1	2A	2B	2C	4A	4B	4C	4D	5	8A	8B	8C	8D	8E	8F	10	16A	16B	16C	16D	16E	16F	16G	16H	20A	20B	20C	40A	40B
size	1	1	10	40	2	5	5	8	4	2	2	10	10	40	40	4	10	10	10	10	10	10	10	10	8	16	16	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	-i	i	i	i	i	-i	-i	-i	1	-1	-1	1	1	linear of order 4
ρ₆	1	1	-1	1	1	-1	-1	-1	1	1	1	-1	-1	-i	i	1	i	-i	-i	-i	-i	i	i	i	1	-1	-1	1	1	linear of order 4
ρ₇	1	1	-1	-1	1	-1	-1	1	1	1	1	-1	-1	i	-i	1	i	-i	-i	-i	-i	i	i	i	1	1	1	1	1	linear of order 4
ρ₈	1	1	-1	-1	1	-1	-1	1	1	1	1	-1	-1	-i	i	1	-i	i	i	i	i	-i	-i	-i	1	1	1	1	1	linear of order 4
ρ₉	2	2	-2	0	2	-2	-2	0	2	-2	-2	2	2	0	0	2	0	0	0	0	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	2	2	2	0	2	-2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	0	2	0	0	-2	-2	orthogonal lifted from D₄
ρ₁₁	2	2	-2	0	-2	2	2	0	2	0	0	0	0	0	0	2	√2	-√2	-√2	√2	√2	-√2	-√2	√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₂	2	2	-2	0	-2	2	2	0	2	0	0	0	0	0	0	2	-√2	√2	√2	-√2	-√2	√2	√2	-√2	-2	0	0	0	0	orthogonal lifted from D₈
ρ₁₃	2	2	2	0	-2	-2	-2	0	2	0	0	0	0	0	0	2	-√-2	-√-2	-√-2	√-2	√-2	√-2	√-2	-√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₄	2	2	2	0	-2	-2	-2	0	2	0	0	0	0	0	0	2	√-2	√-2	√-2	-√-2	-√-2	-√-2	-√-2	√-2	-2	0	0	0	0	complex lifted from SD₁₆
ρ₁₅	2	-2	0	0	0	2i	-2i	0	2	-√2	√2	√-2	-√-2	0	0	-2	ζ₁₆⁹+ζ₁₆³	ζ₁₆³+ζ₁₆	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	0	0	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	ζ₁₆³+ζ₁₆	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁹+ζ₁₆³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆⁹	0	0	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	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆³+ζ₁₆	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁹+ζ₁₆³	ζ₁₆⁷+ζ₁₆⁵	0	0	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	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆³+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁹+ζ₁₆³	0	0	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	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆³+ζ₁₆	ζ₁₆¹³+ζ₁₆⁷	0	0	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	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆³+ζ₁₆	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	0	0	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	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹¹+ζ₁₆	ζ₁₆⁹+ζ₁₆³	ζ₁₆³+ζ₁₆	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆¹⁵+ζ₁₆⁵	0	0	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	ζ₁₆¹¹+ζ₁₆⁹	ζ₁₆⁹+ζ₁₆³	ζ₁₆¹¹+ζ₁₆	ζ₁₆¹⁵+ζ₁₆¹³	ζ₁₆⁷+ζ₁₆⁵	ζ₁₆¹³+ζ₁₆⁷	ζ₁₆¹⁵+ζ₁₆⁵	ζ₁₆³+ζ₁₆	0	0	0	-√2	√2	complex lifted from D₈.C₄
ρ₂₃	4	4	0	0	4	0	0	4	-1	4	4	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	-1	-1	-1	-1	orthogonal lifted from F₅
ρ₂₄	4	4	0	0	4	0	0	-4	-1	4	4	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	1	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	0	0	0	0	0	0	0	0	-1	-√5	√5	1	1	orthogonal lifted from C₂²⋊F₅
ρ₂₆	4	4	0	0	4	0	0	0	-1	-4	-4	0	0	0	0	-1	0	0	0	0	0	0	0	0	-1	√5	-√5	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	2	0	0	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	orthogonal 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	orthogonal faithful, Schur index 2

Smallest permutation representation of Q₁₆.F₅
►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 87 5 83)(2 86 6 82)(3 85 7 81)(4 84 8 88)(9 120 13 116)(10 119 14 115)(11 118 15 114)(12 117 16 113)(17 135 21 131)(18 134 22 130)(19 133 23 129)(20 132 24 136)(25 139 29 143)(26 138 30 142)(27 137 31 141)(28 144 32 140)(33 127 37 123)(34 126 38 122)(35 125 39 121)(36 124 40 128)(41 106 45 110)(42 105 46 109)(43 112 47 108)(44 111 48 107)(49 93 53 89)(50 92 54 96)(51 91 55 95)(52 90 56 94)(57 97 61 101)(58 104 62 100)(59 103 63 99)(60 102 64 98)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)(145 159 149 155)(146 158 150 154)(147 157 151 153)(148 156 152 160)

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

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

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

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

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

Matrix representation of Q₁₆.F₅ ►in GL₆(𝔽₂₄₁)

11	230	0	0	0	0
11	11	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

64	0	0	0	0	0
0	177	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

87	140	0	0	0	0
140	154	0	0	0	0
0	0	58	116	14	102
0	0	139	227	125	183
0	0	139	197	14	153
0	0	58	197	44	183

G:=sub<GL(6,GF(241))| [11,11,0,0,0,0,230,11,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[64,0,0,0,0,0,0,177,0,0,0,0,0,0,117,7,0,234,0,0,0,124,7,234,0,0,234,7,124,0,0,0,234,0,7,117],[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],[87,140,0,0,0,0,140,154,0,0,0,0,0,0,58,139,139,58,0,0,116,227,197,197,0,0,14,125,14,44,0,0,102,183,153,183] >;

Q₁₆.F₅ in GAP, Magma, Sage, TeX

Q_{16}.F_5

% in TeX

G:=Group("Q16.F5");

// GroupNames label

G:=SmallGroup(320,247);

// by ID

G=gap.SmallGroup(320,247);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,184,675,346,192,1684,851,102,6278,3156]);

// Polycyclic

G:=Group<a,b,c,d|a^8=c^5=1,b^2=d^4=a^4,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;

// generators/relations

Export

Subgroup lattice of Q₁₆.F₅ in TeX
Character table of Q₁₆.F₅ in TeX

87	140	0	0	0	0
140	154	0	0	0	0
0	0	58	116	14	102
0	0	139	227	125	183
0	0	139	197	14	153
0	0	58	197	44	183

87	140	0	0	0	0
140	154	0	0	0	0
0	0	58	116	14	102
0	0	139	227	125	183
0	0	139	197	14	153
0	0	58	197	44	183

G = Q16.F5 order 320 = 26·5

1st non-split extension by Q16 of F5 acting via F5/D5=C2

G = Q₁₆.F₅ order 320 = 2⁶·5

1^st non-split extension by Q₁₆ of F₅ acting via F₅/D₅=C₂

87	140	0	0	0	0
140	154	0	0	0	0
0	0	58	116	14	102
0	0	139	227	125	183
0	0	139	197	14	153
0	0	58	197	44	183