direct product, non-abelian, soluble
Aliases: C10×C4.A4, C4○D4⋊2C30, C4.6(C10×A4), (C2×C20).2A4, C20.12(C2×A4), (C2×Q8).2C30, Q8.1(C2×C30), (Q8×C10).4C6, C22.9(C10×A4), C10.16(C22×A4), SL2(𝔽3)⋊3(C2×C10), (C2×SL2(𝔽3))⋊4C10, (C10×SL2(𝔽3))⋊9C2, (C5×SL2(𝔽3))⋊11C22, (C10×C4○D4)⋊C3, (C2×C4○D4)⋊C15, C2.5(A4×C2×C10), (C5×C4○D4)⋊4C6, (C2×C4).2(C5×A4), (C2×C10).18(C2×A4), (C5×Q8).11(C2×C6), SmallGroup(480,1130)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — C10×C4.A4 |
| Q8 — C10×C4.A4 |
Subgroups: 262 in 98 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6 [×3], C2×C4, C2×C4 [×5], D4 [×4], Q8, Q8, C23, C10, C10 [×2], C10 [×2], C12 [×2], C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4 [×2], C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], SL2(𝔽3), C2×C12, C30 [×3], C2×C4○D4, C2×C20, C2×C20 [×5], C5×D4 [×4], C5×Q8, C5×Q8, C22×C10, C2×SL2(𝔽3), C4.A4 [×2], C60 [×2], C2×C30, C22×C20, D4×C10, Q8×C10, C5×C4○D4 [×2], C5×C4○D4 [×2], C2×C4.A4, C5×SL2(𝔽3), C2×C60, C10×C4○D4, C10×SL2(𝔽3), C5×C4.A4 [×2], C10×C4.A4
Quotients:
C1, C2 [×3], C3, C22, C5, C6 [×3], C10 [×3], A4, C2×C6, C15, C2×C10, C2×A4 [×3], C30 [×3], C4.A4 [×2], C22×A4, C5×A4, C2×C30, C2×C4.A4, C10×A4 [×3], C5×C4.A4 [×2], A4×C2×C10, C10×C4.A4
Generators and relations
G = < a,b,c,d,e | a10=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >
►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 82 121 44)(2 83 122 45)(3 84 123 46)(4 85 124 47)(5 86 125 48)(6 87 126 49)(7 88 127 50)(8 89 128 41)(9 90 129 42)(10 81 130 43)(11 32 135 153)(12 33 136 154)(13 34 137 155)(14 35 138 156)(15 36 139 157)(16 37 140 158)(17 38 131 159)(18 39 132 160)(19 40 133 151)(20 31 134 152)(21 99 145 51)(22 100 146 52)(23 91 147 53)(24 92 148 54)(25 93 149 55)(26 94 150 56)(27 95 141 57)(28 96 142 58)(29 97 143 59)(30 98 144 60)(61 74 109 112)(62 75 110 113)(63 76 101 114)(64 77 102 115)(65 78 103 116)(66 79 104 117)(67 80 105 118)(68 71 106 119)(69 72 107 120)(70 73 108 111)
(1 140 121 16)(2 131 122 17)(3 132 123 18)(4 133 124 19)(5 134 125 20)(6 135 126 11)(7 136 127 12)(8 137 128 13)(9 138 129 14)(10 139 130 15)(21 62 145 110)(22 63 146 101)(23 64 147 102)(24 65 148 103)(25 66 149 104)(26 67 150 105)(27 68 141 106)(28 69 142 107)(29 70 143 108)(30 61 144 109)(31 86 152 48)(32 87 153 49)(33 88 154 50)(34 89 155 41)(35 90 156 42)(36 81 157 43)(37 82 158 44)(38 83 159 45)(39 84 160 46)(40 85 151 47)(51 113 99 75)(52 114 100 76)(53 115 91 77)(54 116 92 78)(55 117 93 79)(56 118 94 80)(57 119 95 71)(58 120 96 72)(59 111 97 73)(60 112 98 74)
(1 69 121 107)(2 70 122 108)(3 61 123 109)(4 62 124 110)(5 63 125 101)(6 64 126 102)(7 65 127 103)(8 66 128 104)(9 67 129 105)(10 68 130 106)(11 147 135 23)(12 148 136 24)(13 149 137 25)(14 150 138 26)(15 141 139 27)(16 142 140 28)(17 143 131 29)(18 144 132 30)(19 145 133 21)(20 146 134 22)(31 52 152 100)(32 53 153 91)(33 54 154 92)(34 55 155 93)(35 56 156 94)(36 57 157 95)(37 58 158 96)(38 59 159 97)(39 60 160 98)(40 51 151 99)(41 117 89 79)(42 118 90 80)(43 119 81 71)(44 120 82 72)(45 111 83 73)(46 112 84 74)(47 113 85 75)(48 114 86 76)(49 115 87 77)(50 116 88 78)
(11 102 23)(12 103 24)(13 104 25)(14 105 26)(15 106 27)(16 107 28)(17 108 29)(18 109 30)(19 110 21)(20 101 22)(31 114 100)(32 115 91)(33 116 92)(34 117 93)(35 118 94)(36 119 95)(37 120 96)(38 111 97)(39 112 98)(40 113 99)(51 151 75)(52 152 76)(53 153 77)(54 154 78)(55 155 79)(56 156 80)(57 157 71)(58 158 72)(59 159 73)(60 160 74)(61 144 132)(62 145 133)(63 146 134)(64 147 135)(65 148 136)(66 149 137)(67 150 138)(68 141 139)(69 142 140)(70 143 131)
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), (1,82,121,44)(2,83,122,45)(3,84,123,46)(4,85,124,47)(5,86,125,48)(6,87,126,49)(7,88,127,50)(8,89,128,41)(9,90,129,42)(10,81,130,43)(11,32,135,153)(12,33,136,154)(13,34,137,155)(14,35,138,156)(15,36,139,157)(16,37,140,158)(17,38,131,159)(18,39,132,160)(19,40,133,151)(20,31,134,152)(21,99,145,51)(22,100,146,52)(23,91,147,53)(24,92,148,54)(25,93,149,55)(26,94,150,56)(27,95,141,57)(28,96,142,58)(29,97,143,59)(30,98,144,60)(61,74,109,112)(62,75,110,113)(63,76,101,114)(64,77,102,115)(65,78,103,116)(66,79,104,117)(67,80,105,118)(68,71,106,119)(69,72,107,120)(70,73,108,111), (1,140,121,16)(2,131,122,17)(3,132,123,18)(4,133,124,19)(5,134,125,20)(6,135,126,11)(7,136,127,12)(8,137,128,13)(9,138,129,14)(10,139,130,15)(21,62,145,110)(22,63,146,101)(23,64,147,102)(24,65,148,103)(25,66,149,104)(26,67,150,105)(27,68,141,106)(28,69,142,107)(29,70,143,108)(30,61,144,109)(31,86,152,48)(32,87,153,49)(33,88,154,50)(34,89,155,41)(35,90,156,42)(36,81,157,43)(37,82,158,44)(38,83,159,45)(39,84,160,46)(40,85,151,47)(51,113,99,75)(52,114,100,76)(53,115,91,77)(54,116,92,78)(55,117,93,79)(56,118,94,80)(57,119,95,71)(58,120,96,72)(59,111,97,73)(60,112,98,74), (1,69,121,107)(2,70,122,108)(3,61,123,109)(4,62,124,110)(5,63,125,101)(6,64,126,102)(7,65,127,103)(8,66,128,104)(9,67,129,105)(10,68,130,106)(11,147,135,23)(12,148,136,24)(13,149,137,25)(14,150,138,26)(15,141,139,27)(16,142,140,28)(17,143,131,29)(18,144,132,30)(19,145,133,21)(20,146,134,22)(31,52,152,100)(32,53,153,91)(33,54,154,92)(34,55,155,93)(35,56,156,94)(36,57,157,95)(37,58,158,96)(38,59,159,97)(39,60,160,98)(40,51,151,99)(41,117,89,79)(42,118,90,80)(43,119,81,71)(44,120,82,72)(45,111,83,73)(46,112,84,74)(47,113,85,75)(48,114,86,76)(49,115,87,77)(50,116,88,78), (11,102,23)(12,103,24)(13,104,25)(14,105,26)(15,106,27)(16,107,28)(17,108,29)(18,109,30)(19,110,21)(20,101,22)(31,114,100)(32,115,91)(33,116,92)(34,117,93)(35,118,94)(36,119,95)(37,120,96)(38,111,97)(39,112,98)(40,113,99)(51,151,75)(52,152,76)(53,153,77)(54,154,78)(55,155,79)(56,156,80)(57,157,71)(58,158,72)(59,159,73)(60,160,74)(61,144,132)(62,145,133)(63,146,134)(64,147,135)(65,148,136)(66,149,137)(67,150,138)(68,141,139)(69,142,140)(70,143,131)>;
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), (1,82,121,44)(2,83,122,45)(3,84,123,46)(4,85,124,47)(5,86,125,48)(6,87,126,49)(7,88,127,50)(8,89,128,41)(9,90,129,42)(10,81,130,43)(11,32,135,153)(12,33,136,154)(13,34,137,155)(14,35,138,156)(15,36,139,157)(16,37,140,158)(17,38,131,159)(18,39,132,160)(19,40,133,151)(20,31,134,152)(21,99,145,51)(22,100,146,52)(23,91,147,53)(24,92,148,54)(25,93,149,55)(26,94,150,56)(27,95,141,57)(28,96,142,58)(29,97,143,59)(30,98,144,60)(61,74,109,112)(62,75,110,113)(63,76,101,114)(64,77,102,115)(65,78,103,116)(66,79,104,117)(67,80,105,118)(68,71,106,119)(69,72,107,120)(70,73,108,111), (1,140,121,16)(2,131,122,17)(3,132,123,18)(4,133,124,19)(5,134,125,20)(6,135,126,11)(7,136,127,12)(8,137,128,13)(9,138,129,14)(10,139,130,15)(21,62,145,110)(22,63,146,101)(23,64,147,102)(24,65,148,103)(25,66,149,104)(26,67,150,105)(27,68,141,106)(28,69,142,107)(29,70,143,108)(30,61,144,109)(31,86,152,48)(32,87,153,49)(33,88,154,50)(34,89,155,41)(35,90,156,42)(36,81,157,43)(37,82,158,44)(38,83,159,45)(39,84,160,46)(40,85,151,47)(51,113,99,75)(52,114,100,76)(53,115,91,77)(54,116,92,78)(55,117,93,79)(56,118,94,80)(57,119,95,71)(58,120,96,72)(59,111,97,73)(60,112,98,74), (1,69,121,107)(2,70,122,108)(3,61,123,109)(4,62,124,110)(5,63,125,101)(6,64,126,102)(7,65,127,103)(8,66,128,104)(9,67,129,105)(10,68,130,106)(11,147,135,23)(12,148,136,24)(13,149,137,25)(14,150,138,26)(15,141,139,27)(16,142,140,28)(17,143,131,29)(18,144,132,30)(19,145,133,21)(20,146,134,22)(31,52,152,100)(32,53,153,91)(33,54,154,92)(34,55,155,93)(35,56,156,94)(36,57,157,95)(37,58,158,96)(38,59,159,97)(39,60,160,98)(40,51,151,99)(41,117,89,79)(42,118,90,80)(43,119,81,71)(44,120,82,72)(45,111,83,73)(46,112,84,74)(47,113,85,75)(48,114,86,76)(49,115,87,77)(50,116,88,78), (11,102,23)(12,103,24)(13,104,25)(14,105,26)(15,106,27)(16,107,28)(17,108,29)(18,109,30)(19,110,21)(20,101,22)(31,114,100)(32,115,91)(33,116,92)(34,117,93)(35,118,94)(36,119,95)(37,120,96)(38,111,97)(39,112,98)(40,113,99)(51,151,75)(52,152,76)(53,153,77)(54,154,78)(55,155,79)(56,156,80)(57,157,71)(58,158,72)(59,159,73)(60,160,74)(61,144,132)(62,145,133)(63,146,134)(64,147,135)(65,148,136)(66,149,137)(67,150,138)(68,141,139)(69,142,140)(70,143,131) );
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)], [(1,82,121,44),(2,83,122,45),(3,84,123,46),(4,85,124,47),(5,86,125,48),(6,87,126,49),(7,88,127,50),(8,89,128,41),(9,90,129,42),(10,81,130,43),(11,32,135,153),(12,33,136,154),(13,34,137,155),(14,35,138,156),(15,36,139,157),(16,37,140,158),(17,38,131,159),(18,39,132,160),(19,40,133,151),(20,31,134,152),(21,99,145,51),(22,100,146,52),(23,91,147,53),(24,92,148,54),(25,93,149,55),(26,94,150,56),(27,95,141,57),(28,96,142,58),(29,97,143,59),(30,98,144,60),(61,74,109,112),(62,75,110,113),(63,76,101,114),(64,77,102,115),(65,78,103,116),(66,79,104,117),(67,80,105,118),(68,71,106,119),(69,72,107,120),(70,73,108,111)], [(1,140,121,16),(2,131,122,17),(3,132,123,18),(4,133,124,19),(5,134,125,20),(6,135,126,11),(7,136,127,12),(8,137,128,13),(9,138,129,14),(10,139,130,15),(21,62,145,110),(22,63,146,101),(23,64,147,102),(24,65,148,103),(25,66,149,104),(26,67,150,105),(27,68,141,106),(28,69,142,107),(29,70,143,108),(30,61,144,109),(31,86,152,48),(32,87,153,49),(33,88,154,50),(34,89,155,41),(35,90,156,42),(36,81,157,43),(37,82,158,44),(38,83,159,45),(39,84,160,46),(40,85,151,47),(51,113,99,75),(52,114,100,76),(53,115,91,77),(54,116,92,78),(55,117,93,79),(56,118,94,80),(57,119,95,71),(58,120,96,72),(59,111,97,73),(60,112,98,74)], [(1,69,121,107),(2,70,122,108),(3,61,123,109),(4,62,124,110),(5,63,125,101),(6,64,126,102),(7,65,127,103),(8,66,128,104),(9,67,129,105),(10,68,130,106),(11,147,135,23),(12,148,136,24),(13,149,137,25),(14,150,138,26),(15,141,139,27),(16,142,140,28),(17,143,131,29),(18,144,132,30),(19,145,133,21),(20,146,134,22),(31,52,152,100),(32,53,153,91),(33,54,154,92),(34,55,155,93),(35,56,156,94),(36,57,157,95),(37,58,158,96),(38,59,159,97),(39,60,160,98),(40,51,151,99),(41,117,89,79),(42,118,90,80),(43,119,81,71),(44,120,82,72),(45,111,83,73),(46,112,84,74),(47,113,85,75),(48,114,86,76),(49,115,87,77),(50,116,88,78)], [(11,102,23),(12,103,24),(13,104,25),(14,105,26),(15,106,27),(16,107,28),(17,108,29),(18,109,30),(19,110,21),(20,101,22),(31,114,100),(32,115,91),(33,116,92),(34,117,93),(35,118,94),(36,119,95),(37,120,96),(38,111,97),(39,112,98),(40,113,99),(51,151,75),(52,152,76),(53,153,77),(54,154,78),(55,155,79),(56,156,80),(57,157,71),(58,158,72),(59,159,73),(60,160,74),(61,144,132),(62,145,133),(63,146,134),(64,147,135),(65,148,136),(66,149,137),(67,150,138),(68,141,139),(69,142,140),(70,143,131)])
Matrix representation ►G ⊆ GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 1 | 59 |
| 0 | 1 | 60 |
| 1 | 0 | 0 |
| 0 | 34 | 28 |
| 0 | 48 | 27 |
| 47 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 13 |
G:=sub<GL(3,GF(61))| [60,0,0,0,3,0,0,0,3],[1,0,0,0,11,0,0,0,11],[1,0,0,0,1,1,0,59,60],[1,0,0,0,34,48,0,28,27],[47,0,0,0,1,1,0,0,13] >;
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12H | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20X | 30A | ··· | 30X | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 4 | 4 | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 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 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | C4.A4 | C5×C4.A4 | A4 | C2×A4 | C2×A4 | C5×A4 | C10×A4 | C10×A4 |
| kernel | C10×C4.A4 | C10×SL2(𝔽3) | C5×C4.A4 | C10×C4○D4 | C2×C4.A4 | Q8×C10 | C5×C4○D4 | C2×SL2(𝔽3) | C4.A4 | C2×C4○D4 | C2×Q8 | C4○D4 | C10 | C2 | C2×C20 | C20 | C2×C10 | C2×C4 | C4 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 12 | 48 | 1 | 2 | 1 | 4 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_{10}\times C_4.A_4 % in TeX
G:=Group("C10xC4.A4"); // GroupNames label
G:=SmallGroup(480,1130);
// by ID
G=gap.SmallGroup(480,1130);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1688,1068,172,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=e^3=1,c^2=d^2=b^2,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^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀