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G = D8.F5order 320 = 26·5

1st non-split extension by D8 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8.1F5, Dic20.2C4, D10.5SD16, Dic5.21D8, D5⋊C162C2, C40.9(C2×C4), (C5×D8).2C4, C8.11(C2×F5), (C4×D5).22D4, C52C8.13D4, D10.Q81C2, D83D5.4C2, C51(D8.C4), C4.3(C22⋊F5), C20.3(C22⋊C4), C2.8(D20⋊C4), (C8×D5).18C22, C10.7(D4⋊C4), SmallGroup(320,243)

Series: Derived Chief Lower central Upper central

C1C40 — D8.F5
C1C5C10C20C4×D5C8×D5D10.Q8 — D8.F5
C5C10C20C40 — D8.F5
C1C2C4C8D8

Generators and relations for D8.F5
 G = < a,b,c,d | a8=b2=c5=1, d4=a4, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=c3 >

8C2
10C2
4C22
5C4
5C22
20C4
2D5
8C10
2D4
5C2×C4
5C8
10Q8
20C2×C4
20D4
20C8
4C2×C10
4Dic5
5Q16
5C2×C8
10C4○D4
10SD16
10M4(2)
10C16
2C5×D4
2Dic10
4C2×Dic5
4C5⋊D4
4C5⋊C8
5C4○D8
5C2×C16
5C8.C4
2C5⋊C16
2C4.F5
2D42D5
2D4.D5
5D8.C4

Character table of D8.F5

 class 12A2B2C4A4B4C4D58A8B8C8D8E8F10A10B10C16A16B16C16D16E16F16G16H2040A40B
 size 118102554042210104040416161010101010101010888
ρ111111111111111111111111111111    trivial
ρ211-11111-111111-1-11-1-111111111111    linear of order 2
ρ311-11111-111111111-1-1-1-1-1-1-1-1-1-1111    linear of order 2
ρ41111111111111-1-1111-1-1-1-1-1-1-1-1111    linear of order 2
ρ5111-11-1-1-1111-1-1i-i111i-i-i-i-iiii111    linear of order 4
ρ611-1-11-1-11111-1-1i-i1-1-1-iiiii-i-i-i111    linear of order 4
ρ711-1-11-1-11111-1-1-ii1-1-1i-i-i-i-iiii111    linear of order 4
ρ8111-11-1-1-1111-1-1-ii111-iiiii-i-i-i111    linear of order 4
ρ9220-22-2-202-2-22200200000000002-2-2    orthogonal lifted from D4
ρ10220222202-2-2-2-200200000000002-2-2    orthogonal lifted from D4
ρ11220-2-22202000000200-222-2-222-2-200    orthogonal lifted from D8
ρ12220-2-222020000002002-2-222-2-22-200    orthogonal lifted from D8
ρ132202-2-2-202000000200--2--2--2-2-2-2-2--2-200    complex lifted from SD16
ρ142202-2-2-202000000200-2-2-2--2--2--2--2-2-200    complex lifted from SD16
ρ152-2000-2i2i02-22-2--200-200ζ1613167ζ167165ζ16151613ζ169163ζ161116ζ16316ζ1611169ζ161516502-2    complex lifted from D8.C4
ρ162-20002i-2i02-22--2-200-200ζ161116ζ16316ζ1611169ζ1615165ζ1613167ζ167165ζ16151613ζ16916302-2    complex lifted from D8.C4
ρ172-20002i-2i022-2-2--200-200ζ16151613ζ1613167ζ1615165ζ16316ζ1611169ζ161116ζ169163ζ1671650-22    complex lifted from D8.C4
ρ182-2000-2i2i02-22-2--200-200ζ1615165ζ16151613ζ167165ζ161116ζ169163ζ1611169ζ16316ζ161316702-2    complex lifted from D8.C4
ρ192-2000-2i2i022-2--2-200-200ζ16316ζ169163ζ161116ζ16151613ζ167165ζ1615165ζ1613167ζ16111690-22    complex lifted from D8.C4
ρ202-20002i-2i02-22--2-200-200ζ169163ζ1611169ζ16316ζ1613167ζ1615165ζ16151613ζ167165ζ16111602-2    complex lifted from D8.C4
ρ212-20002i-2i022-2-2--200-200ζ167165ζ1615165ζ1613167ζ1611169ζ16316ζ169163ζ161116ζ161516130-22    complex lifted from D8.C4
ρ222-2000-2i2i022-2--2-200-200ζ1611169ζ161116ζ169163ζ167165ζ16151613ζ1613167ζ1615165ζ163160-22    complex lifted from D8.C4
ρ2344404000-1440000-1-1-100000000-1-1-1    orthogonal lifted from F5
ρ2444-404000-1440000-11100000000-1-1-1    orthogonal lifted from C2×F5
ρ2544004000-1-4-40000-15-500000000-111    orthogonal lifted from C22⋊F5
ρ2644004000-1-4-40000-1-5500000000-111    orthogonal lifted from C22⋊F5
ρ278800-8000-2000000-20000000000200    orthogonal lifted from D20⋊C4, Schur index 2
ρ288-8000000-2-42420000200000000000-22    symplectic faithful, Schur index 2
ρ298-8000000-242-4200002000000000002-2    symplectic faithful, Schur index 2

Smallest permutation representation of D8.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 82)(2 81)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 147)(10 146)(11 145)(12 152)(13 151)(14 150)(15 149)(16 148)(17 132)(18 131)(19 130)(20 129)(21 136)(22 135)(23 134)(24 133)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 128)(40 127)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 80)(48 79)(49 90)(50 89)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 102)(58 101)(59 100)(60 99)(61 98)(62 97)(63 104)(64 103)(65 112)(66 111)(67 110)(68 109)(69 108)(70 107)(71 106)(72 105)(113 157)(114 156)(115 155)(116 154)(117 153)(118 160)(119 159)(120 158)
(1 72 44 61 53)(2 65 45 62 54)(3 66 46 63 55)(4 67 47 64 56)(5 68 48 57 49)(6 69 41 58 50)(7 70 42 59 51)(8 71 43 60 52)(9 24 40 158 28)(10 17 33 159 29)(11 18 34 160 30)(12 19 35 153 31)(13 20 36 154 32)(14 21 37 155 25)(15 22 38 156 26)(16 23 39 157 27)(73 104 92 88 111)(74 97 93 81 112)(75 98 94 82 105)(76 99 95 83 106)(77 100 96 84 107)(78 101 89 85 108)(79 102 90 86 109)(80 103 91 87 110)(113 142 148 134 128)(114 143 149 135 121)(115 144 150 136 122)(116 137 151 129 123)(117 138 152 130 124)(118 139 145 131 125)(119 140 146 132 126)(120 141 147 133 127)
(1 121 3 127 5 125 7 123)(2 128 4 126 6 124 8 122)(9 112 30 91 13 108 26 95)(10 111 31 90 14 107 27 94)(11 110 32 89 15 106 28 93)(12 109 25 96 16 105 29 92)(17 104 153 79 21 100 157 75)(18 103 154 78 22 99 158 74)(19 102 155 77 23 98 159 73)(20 101 156 76 24 97 160 80)(33 88 35 86 37 84 39 82)(34 87 36 85 38 83 40 81)(41 130 60 115 45 134 64 119)(42 129 61 114 46 133 57 118)(43 136 62 113 47 132 58 117)(44 135 63 120 48 131 59 116)(49 145 70 137 53 149 66 141)(50 152 71 144 54 148 67 140)(51 151 72 143 55 147 68 139)(52 150 65 142 56 146 69 138)

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

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

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

Matrix representation of D8.F5 in GL6(𝔽241)

21100000
080000
00240000
00024000
00002400
00000240
,
080000
21100000
001170234234
00712470
00071247
002342340117
,
100000
010000
00240240240240
001000
000100
000010
,
01650000
11500000
0080136136
0013613608
001051131050
00233128128233

G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,0,8,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[0,211,0,0,0,0,8,0,0,0,0,0,0,0,117,7,0,234,0,0,0,124,7,234,0,0,234,7,124,0,0,0,234,0,7,117],[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],[0,115,0,0,0,0,165,0,0,0,0,0,0,0,8,136,105,233,0,0,0,136,113,128,0,0,136,0,105,128,0,0,136,8,0,233] >;

D8.F5 in GAP, Magma, Sage, TeX

D_8.F_5
% in TeX

G:=Group("D8.F5");
// GroupNames label

G:=SmallGroup(320,243);
// by ID

G=gap.SmallGroup(320,243);
# by ID

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

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

Export

Subgroup lattice of D8.F5 in TeX
Character table of D8.F5 in TeX

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