direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42.6C4, C42.12C20, C20.58M4(2), C4⋊C8⋊13C10, C8⋊C4⋊7C10, (C4×C20).27C4, C22⋊C8.7C10, C4.8(C5×M4(2)), (C2×C42).16C10, (C22×C4).13C20, (C22×C20).65C4, C42.60(C2×C10), C23.33(C2×C20), C2.8(C10×M4(2)), C20.351(C4○D4), (C4×C20).301C22, (C2×C40).326C22, (C2×C20).988C23, C10.86(C2×M4(2)), (C2×C10).36M4(2), C22.6(C5×M4(2)), C22.45(C22×C20), C10.79(C42⋊C2), (C22×C20).498C22, (C5×C4⋊C8)⋊32C2, (C2×C4×C20).39C2, (C5×C8⋊C4)⋊21C2, C4.49(C5×C4○D4), (C2×C8).50(C2×C10), (C2×C4).60(C2×C20), (C2×C20).463(C2×C4), (C5×C22⋊C8).16C2, (C22×C4).94(C2×C10), C2.11(C5×C42⋊C2), (C22×C10).187(C2×C4), (C2×C4).156(C22×C10), (C2×C10).339(C22×C4), SmallGroup(320,933)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42.6C4
G = < a,b,c,d | a5=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >
Subgroups: 146 in 110 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C40, C2×C20, C2×C20, C22×C10, C42.6C4, C4×C20, C2×C40, C22×C20, C5×C8⋊C4, C5×C22⋊C8, C5×C4⋊C8, C2×C4×C20, C5×C42.6C4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, M4(2), C22×C4, C4○D4, C20, C2×C10, C42⋊C2, C2×M4(2), C2×C20, C22×C10, C42.6C4, C5×M4(2), C22×C20, C5×C4○D4, C5×C42⋊C2, C10×M4(2), C5×C42.6C4
►On 160 points
Generators in S160
(1 111 31 103 23)(2 112 32 104 24)(3 105 25 97 17)(4 106 26 98 18)(5 107 27 99 19)(6 108 28 100 20)(7 109 29 101 21)(8 110 30 102 22)(9 90 42 114 34)(10 91 43 115 35)(11 92 44 116 36)(12 93 45 117 37)(13 94 46 118 38)(14 95 47 119 39)(15 96 48 120 40)(16 89 41 113 33)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 85 153 73 145)(66 86 154 74 146)(67 87 155 75 147)(68 88 156 76 148)(69 81 157 77 149)(70 82 158 78 150)(71 83 159 79 151)(72 84 160 80 152)
(1 7 5 3)(2 126 6 122)(4 128 8 124)(9 84 13 88)(10 16 14 12)(11 86 15 82)(17 23 21 19)(18 56 22 52)(20 50 24 54)(25 31 29 27)(26 64 30 60)(28 58 32 62)(33 39 37 35)(34 72 38 68)(36 66 40 70)(41 47 45 43)(42 80 46 76)(44 74 48 78)(49 55 53 51)(57 63 61 59)(65 71 69 67)(73 79 77 75)(81 87 85 83)(89 95 93 91)(90 160 94 156)(92 154 96 158)(97 103 101 99)(98 136 102 132)(100 130 104 134)(105 111 109 107)(106 144 110 140)(108 138 112 142)(113 119 117 115)(114 152 118 148)(116 146 120 150)(121 127 125 123)(129 135 133 131)(137 143 141 139)(145 151 149 147)(153 159 157 155)
(1 65 123 37)(2 70 124 34)(3 67 125 39)(4 72 126 36)(5 69 127 33)(6 66 128 38)(7 71 121 35)(8 68 122 40)(9 112 82 140)(10 109 83 137)(11 106 84 142)(12 111 85 139)(13 108 86 144)(14 105 87 141)(15 110 88 138)(16 107 81 143)(17 147 53 119)(18 152 54 116)(19 149 55 113)(20 146 56 118)(21 151 49 115)(22 148 50 120)(23 145 51 117)(24 150 52 114)(25 155 61 95)(26 160 62 92)(27 157 63 89)(28 154 64 94)(29 159 57 91)(30 156 58 96)(31 153 59 93)(32 158 60 90)(41 99 77 135)(42 104 78 132)(43 101 79 129)(44 98 80 134)(45 103 73 131)(46 100 74 136)(47 97 75 133)(48 102 76 130)
(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)
G:=sub<Sym(160)| (1,111,31,103,23)(2,112,32,104,24)(3,105,25,97,17)(4,106,26,98,18)(5,107,27,99,19)(6,108,28,100,20)(7,109,29,101,21)(8,110,30,102,22)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,7,5,3)(2,126,6,122)(4,128,8,124)(9,84,13,88)(10,16,14,12)(11,86,15,82)(17,23,21,19)(18,56,22,52)(20,50,24,54)(25,31,29,27)(26,64,30,60)(28,58,32,62)(33,39,37,35)(34,72,38,68)(36,66,40,70)(41,47,45,43)(42,80,46,76)(44,74,48,78)(49,55,53,51)(57,63,61,59)(65,71,69,67)(73,79,77,75)(81,87,85,83)(89,95,93,91)(90,160,94,156)(92,154,96,158)(97,103,101,99)(98,136,102,132)(100,130,104,134)(105,111,109,107)(106,144,110,140)(108,138,112,142)(113,119,117,115)(114,152,118,148)(116,146,120,150)(121,127,125,123)(129,135,133,131)(137,143,141,139)(145,151,149,147)(153,159,157,155), (1,65,123,37)(2,70,124,34)(3,67,125,39)(4,72,126,36)(5,69,127,33)(6,66,128,38)(7,71,121,35)(8,68,122,40)(9,112,82,140)(10,109,83,137)(11,106,84,142)(12,111,85,139)(13,108,86,144)(14,105,87,141)(15,110,88,138)(16,107,81,143)(17,147,53,119)(18,152,54,116)(19,149,55,113)(20,146,56,118)(21,151,49,115)(22,148,50,120)(23,145,51,117)(24,150,52,114)(25,155,61,95)(26,160,62,92)(27,157,63,89)(28,154,64,94)(29,159,57,91)(30,156,58,96)(31,153,59,93)(32,158,60,90)(41,99,77,135)(42,104,78,132)(43,101,79,129)(44,98,80,134)(45,103,73,131)(46,100,74,136)(47,97,75,133)(48,102,76,130), (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)>;
G:=Group( (1,111,31,103,23)(2,112,32,104,24)(3,105,25,97,17)(4,106,26,98,18)(5,107,27,99,19)(6,108,28,100,20)(7,109,29,101,21)(8,110,30,102,22)(9,90,42,114,34)(10,91,43,115,35)(11,92,44,116,36)(12,93,45,117,37)(13,94,46,118,38)(14,95,47,119,39)(15,96,48,120,40)(16,89,41,113,33)(49,121,137,57,129)(50,122,138,58,130)(51,123,139,59,131)(52,124,140,60,132)(53,125,141,61,133)(54,126,142,62,134)(55,127,143,63,135)(56,128,144,64,136)(65,85,153,73,145)(66,86,154,74,146)(67,87,155,75,147)(68,88,156,76,148)(69,81,157,77,149)(70,82,158,78,150)(71,83,159,79,151)(72,84,160,80,152), (1,7,5,3)(2,126,6,122)(4,128,8,124)(9,84,13,88)(10,16,14,12)(11,86,15,82)(17,23,21,19)(18,56,22,52)(20,50,24,54)(25,31,29,27)(26,64,30,60)(28,58,32,62)(33,39,37,35)(34,72,38,68)(36,66,40,70)(41,47,45,43)(42,80,46,76)(44,74,48,78)(49,55,53,51)(57,63,61,59)(65,71,69,67)(73,79,77,75)(81,87,85,83)(89,95,93,91)(90,160,94,156)(92,154,96,158)(97,103,101,99)(98,136,102,132)(100,130,104,134)(105,111,109,107)(106,144,110,140)(108,138,112,142)(113,119,117,115)(114,152,118,148)(116,146,120,150)(121,127,125,123)(129,135,133,131)(137,143,141,139)(145,151,149,147)(153,159,157,155), (1,65,123,37)(2,70,124,34)(3,67,125,39)(4,72,126,36)(5,69,127,33)(6,66,128,38)(7,71,121,35)(8,68,122,40)(9,112,82,140)(10,109,83,137)(11,106,84,142)(12,111,85,139)(13,108,86,144)(14,105,87,141)(15,110,88,138)(16,107,81,143)(17,147,53,119)(18,152,54,116)(19,149,55,113)(20,146,56,118)(21,151,49,115)(22,148,50,120)(23,145,51,117)(24,150,52,114)(25,155,61,95)(26,160,62,92)(27,157,63,89)(28,154,64,94)(29,159,57,91)(30,156,58,96)(31,153,59,93)(32,158,60,90)(41,99,77,135)(42,104,78,132)(43,101,79,129)(44,98,80,134)(45,103,73,131)(46,100,74,136)(47,97,75,133)(48,102,76,130), (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) );
G=PermutationGroup([[(1,111,31,103,23),(2,112,32,104,24),(3,105,25,97,17),(4,106,26,98,18),(5,107,27,99,19),(6,108,28,100,20),(7,109,29,101,21),(8,110,30,102,22),(9,90,42,114,34),(10,91,43,115,35),(11,92,44,116,36),(12,93,45,117,37),(13,94,46,118,38),(14,95,47,119,39),(15,96,48,120,40),(16,89,41,113,33),(49,121,137,57,129),(50,122,138,58,130),(51,123,139,59,131),(52,124,140,60,132),(53,125,141,61,133),(54,126,142,62,134),(55,127,143,63,135),(56,128,144,64,136),(65,85,153,73,145),(66,86,154,74,146),(67,87,155,75,147),(68,88,156,76,148),(69,81,157,77,149),(70,82,158,78,150),(71,83,159,79,151),(72,84,160,80,152)], [(1,7,5,3),(2,126,6,122),(4,128,8,124),(9,84,13,88),(10,16,14,12),(11,86,15,82),(17,23,21,19),(18,56,22,52),(20,50,24,54),(25,31,29,27),(26,64,30,60),(28,58,32,62),(33,39,37,35),(34,72,38,68),(36,66,40,70),(41,47,45,43),(42,80,46,76),(44,74,48,78),(49,55,53,51),(57,63,61,59),(65,71,69,67),(73,79,77,75),(81,87,85,83),(89,95,93,91),(90,160,94,156),(92,154,96,158),(97,103,101,99),(98,136,102,132),(100,130,104,134),(105,111,109,107),(106,144,110,140),(108,138,112,142),(113,119,117,115),(114,152,118,148),(116,146,120,150),(121,127,125,123),(129,135,133,131),(137,143,141,139),(145,151,149,147),(153,159,157,155)], [(1,65,123,37),(2,70,124,34),(3,67,125,39),(4,72,126,36),(5,69,127,33),(6,66,128,38),(7,71,121,35),(8,68,122,40),(9,112,82,140),(10,109,83,137),(11,106,84,142),(12,111,85,139),(13,108,86,144),(14,105,87,141),(15,110,88,138),(16,107,81,143),(17,147,53,119),(18,152,54,116),(19,149,55,113),(20,146,56,118),(21,151,49,115),(22,148,50,120),(23,145,51,117),(24,150,52,114),(25,155,61,95),(26,160,62,92),(27,157,63,89),(28,154,64,94),(29,159,57,91),(30,156,58,96),(31,153,59,93),(32,158,60,90),(41,99,77,135),(42,104,78,132),(43,101,79,129),(44,98,80,134),(45,103,73,131),(46,100,74,136),(47,97,75,133),(48,102,76,130)], [(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)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | M4(2) | C4○D4 | M4(2) | C5×M4(2) | C5×C4○D4 | C5×M4(2) |
| kernel | C5×C42.6C4 | C5×C8⋊C4 | C5×C22⋊C8 | C5×C4⋊C8 | C2×C4×C20 | C4×C20 | C22×C20 | C42.6C4 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C2×C42 | C42 | C22×C4 | C20 | C20 | C2×C10 | C4 | C4 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 4 | 4 | 4 | 16 | 16 | 16 |
Matrix representation of C5×C42.6C4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 23 | 1 |
| 0 | 32 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 4 | 36 | 0 | 0 |
| 5 | 37 | 0 | 0 |
| 0 | 0 | 23 | 2 |
| 0 | 0 | 22 | 18 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,10,0,0,0,0,10],[9,0,0,0,0,9,0,0,0,0,40,23,0,0,0,1],[0,32,0,0,32,0,0,0,0,0,9,0,0,0,0,9],[4,5,0,0,36,37,0,0,0,0,23,22,0,0,2,18] >;
C5×C42.6C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2._6C_4
% in TeX
G:=Group("C5xC4^2.6C4"); // GroupNames label
G:=SmallGroup(320,933);
// by ID
G=gap.SmallGroup(320,933);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,226,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^2,d*c*d^-1=b^2*c>;
// generators/relations