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
| Q8 — C5×C8.A4 |
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 >
Smallest permutation representation of C5×C8.A4
►On 160 points
Generators in S160
(1 128 39 112 23)(2 121 40 105 24)(3 122 33 106 17)(4 123 34 107 18)(5 124 35 108 19)(6 125 36 109 20)(7 126 37 110 21)(8 127 38 111 22)(9 79 151 63 89)(10 80 152 64 90)(11 73 145 57 91)(12 74 146 58 92)(13 75 147 59 93)(14 76 148 60 94)(15 77 149 61 95)(16 78 150 62 96)(25 97 129 41 113)(26 98 130 42 114)(27 99 131 43 115)(28 100 132 44 116)(29 101 133 45 117)(30 102 134 46 118)(31 103 135 47 119)(32 104 136 48 120)(49 83 154 65 138)(50 84 155 66 139)(51 85 156 67 140)(52 86 157 68 141)(53 87 158 69 142)(54 88 159 70 143)(55 81 160 71 144)(56 82 153 72 137)
(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 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(17 149 21 145)(18 150 22 146)(19 151 23 147)(20 152 24 148)(25 141 29 137)(26 142 30 138)(27 143 31 139)(28 144 32 140)(41 157 45 153)(42 158 46 154)(43 159 47 155)(44 160 48 156)(49 98 53 102)(50 99 54 103)(51 100 55 104)(52 101 56 97)(65 114 69 118)(66 115 70 119)(67 116 71 120)(68 117 72 113)(73 106 77 110)(74 107 78 111)(75 108 79 112)(76 109 80 105)(81 136 85 132)(82 129 86 133)(83 130 87 134)(84 131 88 135)(89 128 93 124)(90 121 94 125)(91 122 95 126)(92 123 96 127)
(1 97 5 101)(2 98 6 102)(3 99 7 103)(4 100 8 104)(9 157 13 153)(10 158 14 154)(11 159 15 155)(12 160 16 156)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)(65 80 69 76)(66 73 70 77)(67 74 71 78)(68 75 72 79)(81 96 85 92)(82 89 86 93)(83 90 87 94)(84 91 88 95)(105 114 109 118)(106 115 110 119)(107 116 111 120)(108 117 112 113)(121 130 125 134)(122 131 126 135)(123 132 127 136)(124 133 128 129)(137 151 141 147)(138 152 142 148)(139 145 143 149)(140 146 144 150)
(9 45 153)(10 46 154)(11 47 155)(12 48 156)(13 41 157)(14 42 158)(15 43 159)(16 44 160)(25 141 147)(26 142 148)(27 143 149)(28 144 150)(29 137 151)(30 138 152)(31 139 145)(32 140 146)(49 64 102)(50 57 103)(51 58 104)(52 59 97)(53 60 98)(54 61 99)(55 62 100)(56 63 101)(65 80 118)(66 73 119)(67 74 120)(68 75 113)(69 76 114)(70 77 115)(71 78 116)(72 79 117)(81 96 132)(82 89 133)(83 90 134)(84 91 135)(85 92 136)(86 93 129)(87 94 130)(88 95 131)
G:=sub<Sym(160)| (1,128,39,112,23)(2,121,40,105,24)(3,122,33,106,17)(4,123,34,107,18)(5,124,35,108,19)(6,125,36,109,20)(7,126,37,110,21)(8,127,38,111,22)(9,79,151,63,89)(10,80,152,64,90)(11,73,145,57,91)(12,74,146,58,92)(13,75,147,59,93)(14,76,148,60,94)(15,77,149,61,95)(16,78,150,62,96)(25,97,129,41,113)(26,98,130,42,114)(27,99,131,43,115)(28,100,132,44,116)(29,101,133,45,117)(30,102,134,46,118)(31,103,135,47,119)(32,104,136,48,120)(49,83,154,65,138)(50,84,155,66,139)(51,85,156,67,140)(52,86,157,68,141)(53,87,158,69,142)(54,88,159,70,143)(55,81,160,71,144)(56,82,153,72,137), (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)(9,39,13,35)(10,40,14,36)(11,33,15,37)(12,34,16,38)(17,149,21,145)(18,150,22,146)(19,151,23,147)(20,152,24,148)(25,141,29,137)(26,142,30,138)(27,143,31,139)(28,144,32,140)(41,157,45,153)(42,158,46,154)(43,159,47,155)(44,160,48,156)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(65,114,69,118)(66,115,70,119)(67,116,71,120)(68,117,72,113)(73,106,77,110)(74,107,78,111)(75,108,79,112)(76,109,80,105)(81,136,85,132)(82,129,86,133)(83,130,87,134)(84,131,88,135)(89,128,93,124)(90,121,94,125)(91,122,95,126)(92,123,96,127), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,157,13,153)(10,158,14,154)(11,159,15,155)(12,160,16,156)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,96,85,92)(82,89,86,93)(83,90,87,94)(84,91,88,95)(105,114,109,118)(106,115,110,119)(107,116,111,120)(108,117,112,113)(121,130,125,134)(122,131,126,135)(123,132,127,136)(124,133,128,129)(137,151,141,147)(138,152,142,148)(139,145,143,149)(140,146,144,150), (9,45,153)(10,46,154)(11,47,155)(12,48,156)(13,41,157)(14,42,158)(15,43,159)(16,44,160)(25,141,147)(26,142,148)(27,143,149)(28,144,150)(29,137,151)(30,138,152)(31,139,145)(32,140,146)(49,64,102)(50,57,103)(51,58,104)(52,59,97)(53,60,98)(54,61,99)(55,62,100)(56,63,101)(65,80,118)(66,73,119)(67,74,120)(68,75,113)(69,76,114)(70,77,115)(71,78,116)(72,79,117)(81,96,132)(82,89,133)(83,90,134)(84,91,135)(85,92,136)(86,93,129)(87,94,130)(88,95,131)>;
G:=Group( (1,128,39,112,23)(2,121,40,105,24)(3,122,33,106,17)(4,123,34,107,18)(5,124,35,108,19)(6,125,36,109,20)(7,126,37,110,21)(8,127,38,111,22)(9,79,151,63,89)(10,80,152,64,90)(11,73,145,57,91)(12,74,146,58,92)(13,75,147,59,93)(14,76,148,60,94)(15,77,149,61,95)(16,78,150,62,96)(25,97,129,41,113)(26,98,130,42,114)(27,99,131,43,115)(28,100,132,44,116)(29,101,133,45,117)(30,102,134,46,118)(31,103,135,47,119)(32,104,136,48,120)(49,83,154,65,138)(50,84,155,66,139)(51,85,156,67,140)(52,86,157,68,141)(53,87,158,69,142)(54,88,159,70,143)(55,81,160,71,144)(56,82,153,72,137), (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)(9,39,13,35)(10,40,14,36)(11,33,15,37)(12,34,16,38)(17,149,21,145)(18,150,22,146)(19,151,23,147)(20,152,24,148)(25,141,29,137)(26,142,30,138)(27,143,31,139)(28,144,32,140)(41,157,45,153)(42,158,46,154)(43,159,47,155)(44,160,48,156)(49,98,53,102)(50,99,54,103)(51,100,55,104)(52,101,56,97)(65,114,69,118)(66,115,70,119)(67,116,71,120)(68,117,72,113)(73,106,77,110)(74,107,78,111)(75,108,79,112)(76,109,80,105)(81,136,85,132)(82,129,86,133)(83,130,87,134)(84,131,88,135)(89,128,93,124)(90,121,94,125)(91,122,95,126)(92,123,96,127), (1,97,5,101)(2,98,6,102)(3,99,7,103)(4,100,8,104)(9,157,13,153)(10,158,14,154)(11,159,15,155)(12,160,16,156)(17,27,21,31)(18,28,22,32)(19,29,23,25)(20,30,24,26)(33,43,37,47)(34,44,38,48)(35,45,39,41)(36,46,40,42)(49,64,53,60)(50,57,54,61)(51,58,55,62)(52,59,56,63)(65,80,69,76)(66,73,70,77)(67,74,71,78)(68,75,72,79)(81,96,85,92)(82,89,86,93)(83,90,87,94)(84,91,88,95)(105,114,109,118)(106,115,110,119)(107,116,111,120)(108,117,112,113)(121,130,125,134)(122,131,126,135)(123,132,127,136)(124,133,128,129)(137,151,141,147)(138,152,142,148)(139,145,143,149)(140,146,144,150), (9,45,153)(10,46,154)(11,47,155)(12,48,156)(13,41,157)(14,42,158)(15,43,159)(16,44,160)(25,141,147)(26,142,148)(27,143,149)(28,144,150)(29,137,151)(30,138,152)(31,139,145)(32,140,146)(49,64,102)(50,57,103)(51,58,104)(52,59,97)(53,60,98)(54,61,99)(55,62,100)(56,63,101)(65,80,118)(66,73,119)(67,74,120)(68,75,113)(69,76,114)(70,77,115)(71,78,116)(72,79,117)(81,96,132)(82,89,133)(83,90,134)(84,91,135)(85,92,136)(86,93,129)(87,94,130)(88,95,131) );
G=PermutationGroup([[(1,128,39,112,23),(2,121,40,105,24),(3,122,33,106,17),(4,123,34,107,18),(5,124,35,108,19),(6,125,36,109,20),(7,126,37,110,21),(8,127,38,111,22),(9,79,151,63,89),(10,80,152,64,90),(11,73,145,57,91),(12,74,146,58,92),(13,75,147,59,93),(14,76,148,60,94),(15,77,149,61,95),(16,78,150,62,96),(25,97,129,41,113),(26,98,130,42,114),(27,99,131,43,115),(28,100,132,44,116),(29,101,133,45,117),(30,102,134,46,118),(31,103,135,47,119),(32,104,136,48,120),(49,83,154,65,138),(50,84,155,66,139),(51,85,156,67,140),(52,86,157,68,141),(53,87,158,69,142),(54,88,159,70,143),(55,81,160,71,144),(56,82,153,72,137)], [(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),(9,39,13,35),(10,40,14,36),(11,33,15,37),(12,34,16,38),(17,149,21,145),(18,150,22,146),(19,151,23,147),(20,152,24,148),(25,141,29,137),(26,142,30,138),(27,143,31,139),(28,144,32,140),(41,157,45,153),(42,158,46,154),(43,159,47,155),(44,160,48,156),(49,98,53,102),(50,99,54,103),(51,100,55,104),(52,101,56,97),(65,114,69,118),(66,115,70,119),(67,116,71,120),(68,117,72,113),(73,106,77,110),(74,107,78,111),(75,108,79,112),(76,109,80,105),(81,136,85,132),(82,129,86,133),(83,130,87,134),(84,131,88,135),(89,128,93,124),(90,121,94,125),(91,122,95,126),(92,123,96,127)], [(1,97,5,101),(2,98,6,102),(3,99,7,103),(4,100,8,104),(9,157,13,153),(10,158,14,154),(11,159,15,155),(12,160,16,156),(17,27,21,31),(18,28,22,32),(19,29,23,25),(20,30,24,26),(33,43,37,47),(34,44,38,48),(35,45,39,41),(36,46,40,42),(49,64,53,60),(50,57,54,61),(51,58,55,62),(52,59,56,63),(65,80,69,76),(66,73,70,77),(67,74,71,78),(68,75,72,79),(81,96,85,92),(82,89,86,93),(83,90,87,94),(84,91,88,95),(105,114,109,118),(106,115,110,119),(107,116,111,120),(108,117,112,113),(121,130,125,134),(122,131,126,135),(123,132,127,136),(124,133,128,129),(137,151,141,147),(138,152,142,148),(139,145,143,149),(140,146,144,150)], [(9,45,153),(10,46,154),(11,47,155),(12,48,156),(13,41,157),(14,42,158),(15,43,159),(16,44,160),(25,141,147),(26,142,148),(27,143,149),(28,144,150),(29,137,151),(30,138,152),(31,139,145),(32,140,146),(49,64,102),(50,57,103),(51,58,104),(52,59,97),(53,60,98),(54,61,99),(55,62,100),(56,63,101),(65,80,118),(66,73,119),(67,74,120),(68,75,113),(69,76,114),(70,77,115),(71,78,116),(72,79,117),(81,96,132),(82,89,133),(83,90,134),(84,91,135),(85,92,136),(86,93,129),(87,94,130),(88,95,131)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | ··· | 24H | 30A | ··· | 30H | 40A | ··· | 40P | 40Q | ··· | 40X | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 4 | 4 | 1 | 1 | 6 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | C8.A4 | C5×C8.A4 | A4 | C2×A4 | C4×A4 | C5×A4 | C10×A4 | A4×C20 |
| kernel | C5×C8.A4 | C5×C4.A4 | C5×C8○D4 | C5×SL2(𝔽3) | C8.A4 | C5×C4○D4 | C4.A4 | C5×Q8 | C8○D4 | SL2(𝔽3) | C4○D4 | Q8 | C5 | C1 | C40 | C20 | C10 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 12 | 48 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×C8.A4 ►in GL2(𝔽41) generated by
| 37 | 0 |
| 0 | 37 |
| 3 | 0 |
| 0 | 3 |
| 40 | 15 |
| 19 | 1 |
| 32 | 12 |
| 0 | 9 |
| 0 | 7 |
| 35 | 40 |
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