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G = C5×C8⋊D4order 320 = 26·5

Direct product of C5 and C8⋊D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C8⋊D4, C4019D4, C81(C5×D4), C2.D812C10, C4.60(D4×C10), C22⋊Q84C10, (C2×SD16)⋊1C10, (C2×C20).327D4, C4⋊D4.4C10, C20.467(C2×D4), D4⋊C417C10, C23.15(C5×D4), Q8⋊C417C10, (C10×SD16)⋊12C2, (C2×M4(2))⋊1C10, C22.92(D4×C10), (C22×C10).33D4, C20.265(C4○D4), (C10×M4(2))⋊11C2, (C2×C40).332C22, (C2×C20).927C23, C10.151(C4⋊D4), C10.137(C8⋊C22), (D4×C10).191C22, (Q8×C10).165C22, C10.137(C8.C22), (C22×C20).425C22, C4⋊C4.8(C2×C10), (C5×C2.D8)⋊27C2, C4.10(C5×C4○D4), (C2×C4).32(C5×D4), (C2×C8).21(C2×C10), C2.20(C5×C4⋊D4), C2.12(C5×C8⋊C22), (C5×C22⋊Q8)⋊31C2, (C2×Q8).9(C2×C10), (C5×D4⋊C4)⋊40C2, (C5×Q8⋊C4)⋊40C2, (C2×D4).14(C2×C10), (C2×C10).648(C2×D4), (C5×C4⋊D4).14C2, C2.12(C5×C8.C22), (C5×C4⋊C4).230C22, (C22×C4).43(C2×C10), (C2×C4).102(C22×C10), SmallGroup(320,969)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C8⋊D4
C1C2C22C2×C4C2×C20D4×C10C10×SD16 — C5×C8⋊D4
C1C2C2×C4 — C5×C8⋊D4
C1C2×C10C22×C20 — C5×C8⋊D4

Generators and relations for C5×C8⋊D4
 G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b3, dcd=c-1 >

Subgroups: 234 in 120 conjugacy classes, 54 normal (50 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C8, C2×C4 [×2], C2×C4 [×5], D4 [×4], Q8 [×2], C23, C23, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C40 [×2], C40, C2×C20 [×2], C2×C20 [×5], C5×D4 [×4], C5×Q8 [×2], C22×C10, C22×C10, C8⋊D4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×C40 [×2], C5×M4(2) [×2], C5×SD16 [×2], C22×C20, D4×C10, D4×C10, Q8×C10, C5×D4⋊C4, C5×Q8⋊C4, C5×C2.D8, C5×C4⋊D4, C5×C22⋊Q8, C10×M4(2), C10×SD16, C5×C8⋊D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C8⋊C22, C8.C22, C5×D4 [×4], C22×C10, C8⋊D4, D4×C10 [×2], C5×C4○D4, C5×C4⋊D4, C5×C8⋊C22, C5×C8.C22, C5×C8⋊D4

Smallest permutation representation of C5×C8⋊D4
On 160 points
Generators in S160
(1 143 61 135 53)(2 144 62 136 54)(3 137 63 129 55)(4 138 64 130 56)(5 139 57 131 49)(6 140 58 132 50)(7 141 59 133 51)(8 142 60 134 52)(9 92 108 17 100)(10 93 109 18 101)(11 94 110 19 102)(12 95 111 20 103)(13 96 112 21 104)(14 89 105 22 97)(15 90 106 23 98)(16 91 107 24 99)(25 41 124 33 116)(26 42 125 34 117)(27 43 126 35 118)(28 44 127 36 119)(29 45 128 37 120)(30 46 121 38 113)(31 47 122 39 114)(32 48 123 40 115)(65 83 155 73 147)(66 84 156 74 148)(67 85 157 75 149)(68 86 158 76 150)(69 87 159 77 151)(70 88 160 78 152)(71 81 153 79 145)(72 82 154 80 146)
(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 69 95 28)(2 68 96 27)(3 67 89 26)(4 66 90 25)(5 65 91 32)(6 72 92 31)(7 71 93 30)(8 70 94 29)(9 114 50 146)(10 113 51 145)(11 120 52 152)(12 119 53 151)(13 118 54 150)(14 117 55 149)(15 116 56 148)(16 115 49 147)(17 122 58 154)(18 121 59 153)(19 128 60 160)(20 127 61 159)(21 126 62 158)(22 125 63 157)(23 124 64 156)(24 123 57 155)(33 130 74 98)(34 129 75 97)(35 136 76 104)(36 135 77 103)(37 134 78 102)(38 133 79 101)(39 132 80 100)(40 131 73 99)(41 138 84 106)(42 137 85 105)(43 144 86 112)(44 143 87 111)(45 142 88 110)(46 141 81 109)(47 140 82 108)(48 139 83 107)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 68)(26 71)(27 66)(28 69)(29 72)(30 67)(31 70)(32 65)(33 76)(34 79)(35 74)(36 77)(37 80)(38 75)(39 78)(40 73)(41 86)(42 81)(43 84)(44 87)(45 82)(46 85)(47 88)(48 83)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(89 93)(90 96)(92 94)(97 101)(98 104)(100 102)(105 109)(106 112)(108 110)(113 149)(114 152)(115 147)(116 150)(117 145)(118 148)(119 151)(120 146)(121 157)(122 160)(123 155)(124 158)(125 153)(126 156)(127 159)(128 154)(129 133)(130 136)(132 134)(137 141)(138 144)(140 142)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B5C5D8A8B8C8D10A···10L10M10N10O10P10Q10R10S10T20A···20H20I20J20K20L20M···20X40A···40P
order1222224444445555888810···10101010101010101020···202020202020···2040···40
size111148224888111144441···1444488882···244448···84···4

80 irreducible representations

dim1111111111111111222222224444
type++++++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4D4D4C4○D4C5×D4C5×D4C5×D4C5×C4○D4C8⋊C22C8.C22C5×C8⋊C22C5×C8.C22
kernelC5×C8⋊D4C5×D4⋊C4C5×Q8⋊C4C5×C2.D8C5×C4⋊D4C5×C22⋊Q8C10×M4(2)C10×SD16C8⋊D4D4⋊C4Q8⋊C4C2.D8C4⋊D4C22⋊Q8C2×M4(2)C2×SD16C40C2×C20C22×C10C20C8C2×C4C23C4C10C10C2C2
# reps1111111144444444211284481144

Matrix representation of C5×C8⋊D4 in GL6(𝔽41)

100000
010000
0037000
0003700
0000370
0000037
,
4000000
0400000
001229401
0012124040
001402912
00112929
,
4020000
4010000
000010
0000040
001000
0004000
,
100000
1400000
001000
0004000
000010
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,12,12,1,1,0,0,29,12,40,1,0,0,40,40,29,29,0,0,1,40,12,29],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

C5×C8⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes D_4
% in TeX

G:=Group("C5xC8:D4");
// GroupNames label

G:=SmallGroup(320,969);
// by ID

G=gap.SmallGroup(320,969);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,7004,172]);
// Polycyclic

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

׿
×
𝔽