direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42.12C4, C42.6C20, C20.62M4(2), (C4×C40)⋊7C2, (C2×C4)⋊4C40, (C4×C8)⋊2C10, C4⋊C8⋊17C10, (C2×C20)⋊14C8, C4.9(C2×C40), (C4×C20).26C4, C20.83(C2×C8), C22⋊C8.9C10, C22.5(C2×C40), C2.3(C22×C40), C23.32(C2×C20), (C22×C20).64C4, (C2×C42).15C10, (C22×C4).16C20, C42.92(C2×C10), C10.56(C22×C8), C4.12(C5×M4(2)), C2.5(C10×M4(2)), C20.350(C4○D4), (C2×C40).360C22, (C2×C20).987C23, (C4×C20).352C22, C10.85(C2×M4(2)), C22.21(C22×C20), C10.78(C42⋊C2), (C22×C20).497C22, (C5×C4⋊C8)⋊36C2, (C2×C4×C20).38C2, C4.48(C5×C4○D4), (C2×C8).64(C2×C10), (C2×C4).78(C2×C20), (C2×C10).51(C2×C8), (C2×C20).512(C2×C4), (C5×C22⋊C8).18C2, C2.4(C5×C42⋊C2), (C22×C4).93(C2×C10), (C2×C10).338(C22×C4), (C2×C4).155(C22×C10), (C22×C10).186(C2×C4), SmallGroup(320,932)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42.12C4
G = < a,b,c,d | a5=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >
Subgroups: 146 in 118 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C40, C2×C20, C2×C20, C2×C20, C22×C10, C42.12C4, C4×C20, C2×C40, C22×C20, C4×C40, C5×C22⋊C8, C5×C4⋊C8, C2×C4×C20, C5×C42.12C4
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, M4(2), C22×C4, C4○D4, C20, C2×C10, C42⋊C2, C22×C8, C2×M4(2), C40, C2×C20, C22×C10, C42.12C4, C2×C40, C5×M4(2), C22×C20, C5×C4○D4, C5×C42⋊C2, C22×C40, C10×M4(2), C5×C42.12C4
►On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 42 114 34 94)(10 43 115 35 95)(11 44 116 36 96)(12 45 117 37 89)(13 46 118 38 90)(14 47 119 39 91)(15 48 120 40 92)(16 41 113 33 93)(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)(9 158 13 154)(10 16 14 12)(11 160 15 156)(17 23 21 19)(18 54 22 50)(20 56 24 52)(25 31 29 27)(26 62 30 58)(28 64 32 60)(33 39 37 35)(34 70 38 66)(36 72 40 68)(41 47 45 43)(42 78 46 74)(44 80 48 76)(49 55 53 51)(57 63 61 59)(65 71 69 67)(73 79 77 75)(81 87 85 83)(82 90 86 94)(84 92 88 96)(89 95 93 91)(97 103 101 99)(98 134 102 130)(100 136 104 132)(105 111 109 107)(106 142 110 138)(108 144 112 140)(113 119 117 115)(114 150 118 146)(116 152 120 148)(121 127 125 123)(129 135 133 131)(137 143 141 139)(145 151 149 147)(153 159 157 155)
(1 65 123 39)(2 66 124 40)(3 67 125 33)(4 68 126 34)(5 69 127 35)(6 70 128 36)(7 71 121 37)(8 72 122 38)(9 28 156 62)(10 29 157 63)(11 30 158 64)(12 31 159 57)(13 32 160 58)(14 25 153 59)(15 26 154 60)(16 27 155 61)(17 145 51 119)(18 146 52 120)(19 147 53 113)(20 148 54 114)(21 149 55 115)(22 150 56 116)(23 151 49 117)(24 152 50 118)(41 99 75 133)(42 100 76 134)(43 101 77 135)(44 102 78 136)(45 103 79 129)(46 104 80 130)(47 97 73 131)(48 98 74 132)(81 143 95 109)(82 144 96 110)(83 137 89 111)(84 138 90 112)(85 139 91 105)(86 140 92 106)(87 141 93 107)(88 142 94 108)
(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)| (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,42,114,34,94)(10,43,115,35,95)(11,44,116,36,96)(12,45,117,37,89)(13,46,118,38,90)(14,47,119,39,91)(15,48,120,40,92)(16,41,113,33,93)(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)(9,158,13,154)(10,16,14,12)(11,160,15,156)(17,23,21,19)(18,54,22,50)(20,56,24,52)(25,31,29,27)(26,62,30,58)(28,64,32,60)(33,39,37,35)(34,70,38,66)(36,72,40,68)(41,47,45,43)(42,78,46,74)(44,80,48,76)(49,55,53,51)(57,63,61,59)(65,71,69,67)(73,79,77,75)(81,87,85,83)(82,90,86,94)(84,92,88,96)(89,95,93,91)(97,103,101,99)(98,134,102,130)(100,136,104,132)(105,111,109,107)(106,142,110,138)(108,144,112,140)(113,119,117,115)(114,150,118,146)(116,152,120,148)(121,127,125,123)(129,135,133,131)(137,143,141,139)(145,151,149,147)(153,159,157,155), (1,65,123,39)(2,66,124,40)(3,67,125,33)(4,68,126,34)(5,69,127,35)(6,70,128,36)(7,71,121,37)(8,72,122,38)(9,28,156,62)(10,29,157,63)(11,30,158,64)(12,31,159,57)(13,32,160,58)(14,25,153,59)(15,26,154,60)(16,27,155,61)(17,145,51,119)(18,146,52,120)(19,147,53,113)(20,148,54,114)(21,149,55,115)(22,150,56,116)(23,151,49,117)(24,152,50,118)(41,99,75,133)(42,100,76,134)(43,101,77,135)(44,102,78,136)(45,103,79,129)(46,104,80,130)(47,97,73,131)(48,98,74,132)(81,143,95,109)(82,144,96,110)(83,137,89,111)(84,138,90,112)(85,139,91,105)(86,140,92,106)(87,141,93,107)(88,142,94,108), (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( (1,105,25,97,17)(2,106,26,98,18)(3,107,27,99,19)(4,108,28,100,20)(5,109,29,101,21)(6,110,30,102,22)(7,111,31,103,23)(8,112,32,104,24)(9,42,114,34,94)(10,43,115,35,95)(11,44,116,36,96)(12,45,117,37,89)(13,46,118,38,90)(14,47,119,39,91)(15,48,120,40,92)(16,41,113,33,93)(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)(9,158,13,154)(10,16,14,12)(11,160,15,156)(17,23,21,19)(18,54,22,50)(20,56,24,52)(25,31,29,27)(26,62,30,58)(28,64,32,60)(33,39,37,35)(34,70,38,66)(36,72,40,68)(41,47,45,43)(42,78,46,74)(44,80,48,76)(49,55,53,51)(57,63,61,59)(65,71,69,67)(73,79,77,75)(81,87,85,83)(82,90,86,94)(84,92,88,96)(89,95,93,91)(97,103,101,99)(98,134,102,130)(100,136,104,132)(105,111,109,107)(106,142,110,138)(108,144,112,140)(113,119,117,115)(114,150,118,146)(116,152,120,148)(121,127,125,123)(129,135,133,131)(137,143,141,139)(145,151,149,147)(153,159,157,155), (1,65,123,39)(2,66,124,40)(3,67,125,33)(4,68,126,34)(5,69,127,35)(6,70,128,36)(7,71,121,37)(8,72,122,38)(9,28,156,62)(10,29,157,63)(11,30,158,64)(12,31,159,57)(13,32,160,58)(14,25,153,59)(15,26,154,60)(16,27,155,61)(17,145,51,119)(18,146,52,120)(19,147,53,113)(20,148,54,114)(21,149,55,115)(22,150,56,116)(23,151,49,117)(24,152,50,118)(41,99,75,133)(42,100,76,134)(43,101,77,135)(44,102,78,136)(45,103,79,129)(46,104,80,130)(47,97,73,131)(48,98,74,132)(81,143,95,109)(82,144,96,110)(83,137,89,111)(84,138,90,112)(85,139,91,105)(86,140,92,106)(87,141,93,107)(88,142,94,108), (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([[(1,105,25,97,17),(2,106,26,98,18),(3,107,27,99,19),(4,108,28,100,20),(5,109,29,101,21),(6,110,30,102,22),(7,111,31,103,23),(8,112,32,104,24),(9,42,114,34,94),(10,43,115,35,95),(11,44,116,36,96),(12,45,117,37,89),(13,46,118,38,90),(14,47,119,39,91),(15,48,120,40,92),(16,41,113,33,93),(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),(9,158,13,154),(10,16,14,12),(11,160,15,156),(17,23,21,19),(18,54,22,50),(20,56,24,52),(25,31,29,27),(26,62,30,58),(28,64,32,60),(33,39,37,35),(34,70,38,66),(36,72,40,68),(41,47,45,43),(42,78,46,74),(44,80,48,76),(49,55,53,51),(57,63,61,59),(65,71,69,67),(73,79,77,75),(81,87,85,83),(82,90,86,94),(84,92,88,96),(89,95,93,91),(97,103,101,99),(98,134,102,130),(100,136,104,132),(105,111,109,107),(106,142,110,138),(108,144,112,140),(113,119,117,115),(114,150,118,146),(116,152,120,148),(121,127,125,123),(129,135,133,131),(137,143,141,139),(145,151,149,147),(153,159,157,155)], [(1,65,123,39),(2,66,124,40),(3,67,125,33),(4,68,126,34),(5,69,127,35),(6,70,128,36),(7,71,121,37),(8,72,122,38),(9,28,156,62),(10,29,157,63),(11,30,158,64),(12,31,159,57),(13,32,160,58),(14,25,153,59),(15,26,154,60),(16,27,155,61),(17,145,51,119),(18,146,52,120),(19,147,53,113),(20,148,54,114),(21,149,55,115),(22,150,56,116),(23,151,49,117),(24,152,50,118),(41,99,75,133),(42,100,76,134),(43,101,77,135),(44,102,78,136),(45,103,79,129),(46,104,80,130),(47,97,73,131),(48,98,74,132),(81,143,95,109),(82,144,96,110),(83,137,89,111),(84,138,90,112),(85,139,91,105),(86,140,92,106),(87,141,93,107),(88,142,94,108)], [(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)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20AV | 20AW | ··· | 20BT | 40A | ··· | 40BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C10 | C20 | C20 | C40 | M4(2) | C4○D4 | C5×M4(2) | C5×C4○D4 |
| kernel | C5×C42.12C4 | C4×C40 | C5×C22⋊C8 | C5×C4⋊C8 | C2×C4×C20 | C4×C20 | C22×C20 | C42.12C4 | C2×C20 | C4×C8 | C22⋊C8 | C4⋊C8 | C2×C42 | C42 | C22×C4 | C2×C4 | C20 | C20 | C4 | C4 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 16 | 8 | 8 | 8 | 4 | 16 | 16 | 64 | 4 | 4 | 16 | 16 |
Matrix representation of C5×C42.12C4 ►in GL3(𝔽41) generated by
| 37 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 9 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 13 | 9 |
| 32 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 38 | 0 | 0 |
| 0 | 14 | 32 |
| 0 | 39 | 27 |
G:=sub<GL(3,GF(41))| [37,0,0,0,1,0,0,0,1],[9,0,0,0,32,13,0,0,9],[32,0,0,0,40,0,0,0,40],[38,0,0,0,14,39,0,32,27] >;
C5×C42.12C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2._{12}C_4 % in TeX
G:=Group("C5xC4^2.12C4"); // GroupNames label
G:=SmallGroup(320,932);
// by ID
G=gap.SmallGroup(320,932);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,226,124]);
// 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,c*d=d*c>;
// generators/relations