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G = Q16.F5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D40.2C4, Q16.1F5, D10.6SD16, Dic5.22D8, D5⋊C163C2, C8.13(C2×F5), C40.11(C2×C4), (C4×D5).26D4, C52C8.17D4, D10.Q82C2, (C5×Q16).2C4, C52(D8.C4), C4.7(C22⋊F5), Q8.D10.4C2, C20.7(C22⋊C4), (C8×D5).20C22, C2.12(D20⋊C4), C10.11(D4⋊C4), SmallGroup(320,247)

Series: Derived Chief Lower central Upper central

C1C40 — Q16.F5
C1C5C10C20C4×D5C8×D5D10.Q8 — Q16.F5
C5C10C20C40 — Q16.F5
C1C2C4C8Q16

Generators and relations for Q16.F5
 G = < a,b,c,d | a8=c5=1, b2=d4=a4, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c3 >

10C2
40C2
4C4
5C22
5C4
20C22
2D5
8D5
2Q8
5C2×C4
5C8
10D4
20C2×C4
20D4
20C8
4C20
4D10
5D8
5C2×C8
10C4○D4
10SD16
10M4(2)
10C16
2C5×Q8
2D20
4C4×D5
4D20
4C5⋊C8
5C4○D8
5C2×C16
5C8.C4
2C5⋊C16
2C4.F5
2Q82D5
2Q8⋊D5
5D8.C4

Character table of Q16.F5

 class 12A2B2C4A4B4C4D58A8B8C8D8E8F1016A16B16C16D16E16F16G16H20A20B20C40A40B
 size 111040255842210104040410101010101010108161688
ρ111111111111111111111111111111    trivial
ρ2111-1111-111111111-1-1-1-1-1-1-1-11-1-111    linear of order 2
ρ3111-1111-111111-1-11111111111-1-111    linear of order 2
ρ41111111111111-1-11-1-1-1-1-1-1-1-111111    linear of order 2
ρ511-111-1-1-1111-1-1i-i1-iiiii-i-i-i1-1-111    linear of order 4
ρ611-111-1-1-1111-1-1-ii1i-i-i-i-iiii1-1-111    linear of order 4
ρ711-1-11-1-11111-1-1i-i1i-i-i-i-iiii11111    linear of order 4
ρ811-1-11-1-11111-1-1-ii1-iiiii-i-i-i11111    linear of order 4
ρ922-202-2-202-2-22200200000000200-2-2    orthogonal lifted from D4
ρ10222022202-2-2-2-200200000000200-2-2    orthogonal lifted from D4
ρ1122-20-2220200000022-2-222-2-22-20000    orthogonal lifted from D8
ρ1222-20-222020000002-222-2-222-2-20000    orthogonal lifted from D8
ρ132220-2-2-2020000002--2--2--2-2-2-2-2--2-20000    complex lifted from SD16
ρ142220-2-2-2020000002-2-2-2--2--2--2--2-2-20000    complex lifted from SD16
ρ152-20002i-2i02-22-2--200-2ζ169163ζ16316ζ1611169ζ1615165ζ1613167ζ16151613ζ167165ζ1611160002-2    complex lifted from D8.C4
ρ162-2000-2i2i022-2-2--200-2ζ16316ζ161116ζ169163ζ167165ζ16151613ζ1615165ζ1613167ζ1611169000-22    complex lifted from D8.C4
ρ172-20002i-2i022-2--2-200-2ζ16151613ζ1615165ζ1613167ζ1611169ζ16316ζ161116ζ169163ζ167165000-22    complex lifted from D8.C4
ρ182-20002i-2i02-22-2--200-2ζ161116ζ1611169ζ16316ζ1613167ζ1615165ζ167165ζ16151613ζ1691630002-2    complex lifted from D8.C4
ρ192-2000-2i2i02-22--2-200-2ζ1615165ζ167165ζ16151613ζ169163ζ161116ζ1611169ζ16316ζ16131670002-2    complex lifted from D8.C4
ρ202-20002i-2i022-2--2-200-2ζ167165ζ1613167ζ1615165ζ16316ζ1611169ζ169163ζ161116ζ16151613000-22    complex lifted from D8.C4
ρ212-2000-2i2i02-22--2-200-2ζ1613167ζ16151613ζ167165ζ161116ζ169163ζ16316ζ1611169ζ16151650002-2    complex lifted from D8.C4
ρ222-2000-2i2i022-2-2--200-2ζ1611169ζ169163ζ161116ζ16151613ζ167165ζ1613167ζ1615165ζ16316000-22    complex lifted from D8.C4
ρ2344004004-1440000-100000000-1-1-1-1-1    orthogonal lifted from F5
ρ244400400-4-1440000-100000000-111-1-1    orthogonal lifted from C2×F5
ρ2544004000-1-4-40000-100000000-1-5511    orthogonal lifted from C22⋊F5
ρ2644004000-1-4-40000-100000000-15-511    orthogonal lifted from C22⋊F5
ρ278800-8000-2000000-20000000020000    orthogonal lifted from D20⋊C4, Schur index 2
ρ288-8000000-2-42420000200000000000-22    orthogonal faithful, Schur index 2
ρ298-8000000-242-4200002000000000002-2    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)
(1 102 5 98)(2 101 6 97)(3 100 7 104)(4 99 8 103)(9 133 13 129)(10 132 14 136)(11 131 15 135)(12 130 16 134)(17 143 21 139)(18 142 22 138)(19 141 23 137)(20 140 24 144)(25 149 29 145)(26 148 30 152)(27 147 31 151)(28 146 32 150)(33 113 37 117)(34 120 38 116)(35 119 39 115)(36 118 40 114)(41 96 45 92)(42 95 46 91)(43 94 47 90)(44 93 48 89)(49 81 53 85)(50 88 54 84)(51 87 55 83)(52 86 56 82)(57 107 61 111)(58 106 62 110)(59 105 63 109)(60 112 64 108)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)(121 158 125 154)(122 157 126 153)(123 156 127 160)(124 155 128 159)
(1 44 52 66 63)(2 45 53 67 64)(3 46 54 68 57)(4 47 55 69 58)(5 48 56 70 59)(6 41 49 71 60)(7 42 50 72 61)(8 43 51 65 62)(9 25 153 24 40)(10 26 154 17 33)(11 27 155 18 34)(12 28 156 19 35)(13 29 157 20 36)(14 30 158 21 37)(15 31 159 22 38)(16 32 160 23 39)(73 106 99 90 83)(74 107 100 91 84)(75 108 101 92 85)(76 109 102 93 86)(77 110 103 94 87)(78 111 104 95 88)(79 112 97 96 81)(80 105 98 89 82)(113 132 148 121 143)(114 133 149 122 144)(115 134 150 123 137)(116 135 151 124 138)(117 136 152 125 139)(118 129 145 126 140)(119 130 146 127 141)(120 131 147 128 142)
(1 23 98 142 5 19 102 138)(2 22 99 141 6 18 103 137)(3 21 100 140 7 17 104 144)(4 20 101 139 8 24 97 143)(9 112 148 47 13 108 152 43)(10 111 149 46 14 107 145 42)(11 110 150 45 15 106 146 41)(12 109 151 44 16 105 147 48)(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)(33 88 122 68 37 84 126 72)(34 87 123 67 38 83 127 71)(35 86 124 66 39 82 128 70)(36 85 125 65 40 81 121 69)(49 155 77 115 53 159 73 119)(50 154 78 114 54 158 74 118)(51 153 79 113 55 157 75 117)(52 160 80 120 56 156 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), (1,102,5,98)(2,101,6,97)(3,100,7,104)(4,99,8,103)(9,133,13,129)(10,132,14,136)(11,131,15,135)(12,130,16,134)(17,143,21,139)(18,142,22,138)(19,141,23,137)(20,140,24,144)(25,149,29,145)(26,148,30,152)(27,147,31,151)(28,146,32,150)(33,113,37,117)(34,120,38,116)(35,119,39,115)(36,118,40,114)(41,96,45,92)(42,95,46,91)(43,94,47,90)(44,93,48,89)(49,81,53,85)(50,88,54,84)(51,87,55,83)(52,86,56,82)(57,107,61,111)(58,106,62,110)(59,105,63,109)(60,112,64,108)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78)(121,158,125,154)(122,157,126,153)(123,156,127,160)(124,155,128,159), (1,44,52,66,63)(2,45,53,67,64)(3,46,54,68,57)(4,47,55,69,58)(5,48,56,70,59)(6,41,49,71,60)(7,42,50,72,61)(8,43,51,65,62)(9,25,153,24,40)(10,26,154,17,33)(11,27,155,18,34)(12,28,156,19,35)(13,29,157,20,36)(14,30,158,21,37)(15,31,159,22,38)(16,32,160,23,39)(73,106,99,90,83)(74,107,100,91,84)(75,108,101,92,85)(76,109,102,93,86)(77,110,103,94,87)(78,111,104,95,88)(79,112,97,96,81)(80,105,98,89,82)(113,132,148,121,143)(114,133,149,122,144)(115,134,150,123,137)(116,135,151,124,138)(117,136,152,125,139)(118,129,145,126,140)(119,130,146,127,141)(120,131,147,128,142), (1,23,98,142,5,19,102,138)(2,22,99,141,6,18,103,137)(3,21,100,140,7,17,104,144)(4,20,101,139,8,24,97,143)(9,112,148,47,13,108,152,43)(10,111,149,46,14,107,145,42)(11,110,150,45,15,106,146,41)(12,109,151,44,16,105,147,48)(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)(33,88,122,68,37,84,126,72)(34,87,123,67,38,83,127,71)(35,86,124,66,39,82,128,70)(36,85,125,65,40,81,121,69)(49,155,77,115,53,159,73,119)(50,154,78,114,54,158,74,118)(51,153,79,113,55,157,75,117)(52,160,80,120,56,156,76,116)>;

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

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

Matrix representation of Q16.F5 in GL6(𝔽241)

112300000
11110000
00240000
00024000
00002400
00000240
,
6400000
01770000
001170234234
00712470
00071247
002342340117
,
100000
010000
00240240240240
001000
000100
000010
,
871400000
1401540000
005811614102
00139227125183
0013919714153
005819744183

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

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