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G = C5×C8.A4order 480 = 25·3·5

Direct product of C5 and C8.A4

direct product, non-abelian, soluble

Aliases: C5×C8.A4, C40.A4, Q8.C60, SL2(𝔽3).C20, C8○D4⋊C15, C8.(C5×A4), C4.5(C10×A4), C10.9(C4×A4), C2.3(A4×C20), C20.11(C2×A4), C4○D4.2C30, C4.A4.3C10, (C5×Q8).4C12, (C5×SL2(𝔽3)).3C4, (C5×C8○D4)⋊C3, (C5×C4.A4).6C2, (C5×C4○D4).4C6, SmallGroup(480,660)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C5×C8.A4
C1C2Q8C4○D4C5×C4○D4C5×C4.A4 — C5×C8.A4
Q8 — C5×C8.A4
C1C40

Generators and relations for C5×C8.A4
 G = < a,b,c,d,e | a5=b8=e3=1, c2=d2=b4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b4c, ece-1=b4cd, ede-1=c >

6C2
4C3
3C22
3C4
4C6
6C10
4C15
3C8
3C2×C4
3D4
4C12
3C2×C10
3C20
4C30
3C2×C8
3M4(2)
4C24
3C5×D4
3C40
3C2×C20
4C60
3C2×C40
3C5×M4(2)
4C120

Smallest permutation representation of C5×C8.A4
On 160 points
Generators in S160
(1 53 45 37 29)(2 54 46 38 30)(3 55 47 39 31)(4 56 48 40 32)(5 49 41 33 25)(6 50 42 34 26)(7 51 43 35 27)(8 52 44 36 28)(9 159 151 143 135)(10 160 152 144 136)(11 153 145 137 129)(12 154 146 138 130)(13 155 147 139 131)(14 156 148 140 132)(15 157 149 141 133)(16 158 150 142 134)(17 128 120 112 104)(18 121 113 105 97)(19 122 114 106 98)(20 123 115 107 99)(21 124 116 108 100)(22 125 117 109 101)(23 126 118 110 102)(24 127 119 111 103)(57 91 83 73 65)(58 92 84 74 66)(59 93 85 75 67)(60 94 86 76 68)(61 95 87 77 69)(62 96 88 78 70)(63 89 81 79 71)(64 90 82 80 72)
(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 59 5 63)(2 60 6 64)(3 61 7 57)(4 62 8 58)(9 22 13 18)(10 23 14 19)(11 24 15 20)(12 17 16 21)(25 71 29 67)(26 72 30 68)(27 65 31 69)(28 66 32 70)(33 79 37 75)(34 80 38 76)(35 73 39 77)(36 74 40 78)(41 81 45 85)(42 82 46 86)(43 83 47 87)(44 84 48 88)(49 89 53 93)(50 90 54 94)(51 91 55 95)(52 92 56 96)(97 135 101 131)(98 136 102 132)(99 129 103 133)(100 130 104 134)(105 143 109 139)(106 144 110 140)(107 137 111 141)(108 138 112 142)(113 151 117 147)(114 152 118 148)(115 145 119 149)(116 146 120 150)(121 159 125 155)(122 160 126 156)(123 153 127 157)(124 154 128 158)
(1 99 5 103)(2 100 6 104)(3 101 7 97)(4 102 8 98)(9 91 13 95)(10 92 14 96)(11 93 15 89)(12 94 16 90)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(25 111 29 107)(26 112 30 108)(27 105 31 109)(28 106 32 110)(33 119 37 115)(34 120 38 116)(35 113 39 117)(36 114 40 118)(41 127 45 123)(42 128 46 124)(43 121 47 125)(44 122 48 126)(57 131 61 135)(58 132 62 136)(59 133 63 129)(60 134 64 130)(65 139 69 143)(66 140 70 144)(67 141 71 137)(68 142 72 138)(73 147 77 151)(74 148 78 152)(75 149 79 145)(76 150 80 146)(81 153 85 157)(82 154 86 158)(83 155 87 159)(84 156 88 160)
(9 91 18)(10 92 19)(11 93 20)(12 94 21)(13 95 22)(14 96 23)(15 89 24)(16 90 17)(57 97 135)(58 98 136)(59 99 129)(60 100 130)(61 101 131)(62 102 132)(63 103 133)(64 104 134)(65 105 143)(66 106 144)(67 107 137)(68 108 138)(69 109 139)(70 110 140)(71 111 141)(72 112 142)(73 113 151)(74 114 152)(75 115 145)(76 116 146)(77 117 147)(78 118 148)(79 119 149)(80 120 150)(81 127 157)(82 128 158)(83 121 159)(84 122 160)(85 123 153)(86 124 154)(87 125 155)(88 126 156)

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

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

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

140 conjugacy classes

class 1 2A2B3A3B4A4B4C5A5B5C5D6A6B8A8B8C8D8E8F10A10B10C10D10E10F10G10H12A12B12C12D15A···15H20A···20H20I20J20K20L24A···24H30A···30H40A···40P40Q···40X60A···60P120A···120AF
order1223344455556688888810101010101010101212121215···1520···202020202024···2430···3040···4040···4060···60120···120
size116441161111441111661111666644444···41···166664···44···41···16···64···44···4

140 irreducible representations

dim11111111111122333333
type++++
imageC1C2C3C4C5C6C10C12C15C20C30C60C8.A4C5×C8.A4A4C2×A4C4×A4C5×A4C10×A4A4×C20
kernelC5×C8.A4C5×C4.A4C5×C8○D4C5×SL2(𝔽3)C8.A4C5×C4○D4C4.A4C5×Q8C8○D4SL2(𝔽3)C4○D4Q8C5C1C40C20C10C8C4C2
# reps11224244888161248112448

Matrix representation of C5×C8.A4 in GL2(𝔽41) generated by

370
037
,
30
03
,
4015
191
,
3212
09
,
07
3540
G:=sub<GL(2,GF(41))| [37,0,0,37],[3,0,0,3],[40,19,15,1],[32,0,12,9],[0,35,7,40] >;

C5×C8.A4 in GAP, Magma, Sage, TeX

C_5\times C_8.A_4
% in TeX

G:=Group("C5xC8.A4");
// GroupNames label

G:=SmallGroup(480,660);
// by ID

G=gap.SmallGroup(480,660);
# by ID

G:=PCGroup([7,-2,-3,-5,-2,-2,2,-2,210,248,2111,172,3792,285,124]);
// Polycyclic

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

Export

Subgroup lattice of C5×C8.A4 in TeX

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