direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10xC42:2C2, (C2xC42):4C10, (C4xC20):49C22, C42:14(C2xC10), C24.14(C2xC10), (C2xC20).712C23, (C2xC10).348C24, (C23xC10).14C22, C22.22(C23xC10), C23.72(C22xC10), (C22xC10).86C23, (C22xC20).510C22, (C2xC4xC20):6C2, (C2xC4:C4):17C10, (C10xC4:C4):44C2, C4:C4:13(C2xC10), (C5xC4:C4):69C22, C2.11(C10xC4oD4), C10.230(C2xC4oD4), C22.34(C5xC4oD4), (C10xC22:C4).32C2, C22:C4.11(C2xC10), (C2xC22:C4).12C10, (C2xC4).16(C22xC10), (C2xC10).234(C4oD4), (C22xC4).102(C2xC10), (C5xC22:C4).145C22, SmallGroup(320,1530)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10xC42:2C2
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
Subgroups: 354 in 246 conjugacy classes, 162 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, C23, C23, C23, C10, C10, C42, C22:C4, C4:C4, C22xC4, C24, C20, C2xC10, C2xC10, C2xC10, C2xC42, C2xC22:C4, C2xC4:C4, C42:2C2, C2xC20, C2xC20, C22xC10, C22xC10, C22xC10, C2xC42:2C2, C4xC20, C5xC22:C4, C5xC4:C4, C22xC20, C23xC10, C2xC4xC20, C10xC22:C4, C10xC4:C4, C5xC42:2C2, C10xC42:2C2
Quotients: C1, C2, C22, C5, C23, C10, C4oD4, C24, C2xC10, C42:2C2, C2xC4oD4, C22xC10, C2xC42:2C2, C5xC4oD4, C23xC10, C5xC42:2C2, C10xC4oD4, C10xC42:2C2
►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)
(1 123 62 147)(2 124 63 148)(3 125 64 149)(4 126 65 150)(5 127 66 141)(6 128 67 142)(7 129 68 143)(8 130 69 144)(9 121 70 145)(10 122 61 146)(11 78 153 100)(12 79 154 91)(13 80 155 92)(14 71 156 93)(15 72 157 94)(16 73 158 95)(17 74 159 96)(18 75 160 97)(19 76 151 98)(20 77 152 99)(21 82 34 106)(22 83 35 107)(23 84 36 108)(24 85 37 109)(25 86 38 110)(26 87 39 101)(27 88 40 102)(28 89 31 103)(29 90 32 104)(30 81 33 105)(41 112 59 134)(42 113 60 135)(43 114 51 136)(44 115 52 137)(45 116 53 138)(46 117 54 139)(47 118 55 140)(48 119 56 131)(49 120 57 132)(50 111 58 133)
(1 79 56 107)(2 80 57 108)(3 71 58 109)(4 72 59 110)(5 73 60 101)(6 74 51 102)(7 75 52 103)(8 76 53 104)(9 77 54 105)(10 78 55 106)(11 118 34 146)(12 119 35 147)(13 120 36 148)(14 111 37 149)(15 112 38 150)(16 113 39 141)(17 114 40 142)(18 115 31 143)(19 116 32 144)(20 117 33 145)(21 122 153 140)(22 123 154 131)(23 124 155 132)(24 125 156 133)(25 126 157 134)(26 127 158 135)(27 128 159 136)(28 129 160 137)(29 130 151 138)(30 121 152 139)(41 86 65 94)(42 87 66 95)(43 88 67 96)(44 89 68 97)(45 90 69 98)(46 81 70 99)(47 82 61 100)(48 83 62 91)(49 84 63 92)(50 85 64 93)
(11 153)(12 154)(13 155)(14 156)(15 157)(16 158)(17 159)(18 160)(19 151)(20 152)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 31)(29 32)(30 33)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)
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), (1,123,62,147)(2,124,63,148)(3,125,64,149)(4,126,65,150)(5,127,66,141)(6,128,67,142)(7,129,68,143)(8,130,69,144)(9,121,70,145)(10,122,61,146)(11,78,153,100)(12,79,154,91)(13,80,155,92)(14,71,156,93)(15,72,157,94)(16,73,158,95)(17,74,159,96)(18,75,160,97)(19,76,151,98)(20,77,152,99)(21,82,34,106)(22,83,35,107)(23,84,36,108)(24,85,37,109)(25,86,38,110)(26,87,39,101)(27,88,40,102)(28,89,31,103)(29,90,32,104)(30,81,33,105)(41,112,59,134)(42,113,60,135)(43,114,51,136)(44,115,52,137)(45,116,53,138)(46,117,54,139)(47,118,55,140)(48,119,56,131)(49,120,57,132)(50,111,58,133), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,118,34,146)(12,119,35,147)(13,120,36,148)(14,111,37,149)(15,112,38,150)(16,113,39,141)(17,114,40,142)(18,115,31,143)(19,116,32,144)(20,117,33,145)(21,122,153,140)(22,123,154,131)(23,124,155,132)(24,125,156,133)(25,126,157,134)(26,127,158,135)(27,128,159,136)(28,129,160,137)(29,130,151,138)(30,121,152,139)(41,86,65,94)(42,87,66,95)(43,88,67,96)(44,89,68,97)(45,90,69,98)(46,81,70,99)(47,82,61,100)(48,83,62,91)(49,84,63,92)(50,85,64,93), (11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138)>;
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), (1,123,62,147)(2,124,63,148)(3,125,64,149)(4,126,65,150)(5,127,66,141)(6,128,67,142)(7,129,68,143)(8,130,69,144)(9,121,70,145)(10,122,61,146)(11,78,153,100)(12,79,154,91)(13,80,155,92)(14,71,156,93)(15,72,157,94)(16,73,158,95)(17,74,159,96)(18,75,160,97)(19,76,151,98)(20,77,152,99)(21,82,34,106)(22,83,35,107)(23,84,36,108)(24,85,37,109)(25,86,38,110)(26,87,39,101)(27,88,40,102)(28,89,31,103)(29,90,32,104)(30,81,33,105)(41,112,59,134)(42,113,60,135)(43,114,51,136)(44,115,52,137)(45,116,53,138)(46,117,54,139)(47,118,55,140)(48,119,56,131)(49,120,57,132)(50,111,58,133), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,118,34,146)(12,119,35,147)(13,120,36,148)(14,111,37,149)(15,112,38,150)(16,113,39,141)(17,114,40,142)(18,115,31,143)(19,116,32,144)(20,117,33,145)(21,122,153,140)(22,123,154,131)(23,124,155,132)(24,125,156,133)(25,126,157,134)(26,127,158,135)(27,128,159,136)(28,129,160,137)(29,130,151,138)(30,121,152,139)(41,86,65,94)(42,87,66,95)(43,88,67,96)(44,89,68,97)(45,90,69,98)(46,81,70,99)(47,82,61,100)(48,83,62,91)(49,84,63,92)(50,85,64,93), (11,153)(12,154)(13,155)(14,156)(15,157)(16,158)(17,159)(18,160)(19,151)(20,152)(21,34)(22,35)(23,36)(24,37)(25,38)(26,39)(27,40)(28,31)(29,32)(30,33)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,149)(112,150)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,139)(122,140)(123,131)(124,132)(125,133)(126,134)(127,135)(128,136)(129,137)(130,138) );
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)], [(1,123,62,147),(2,124,63,148),(3,125,64,149),(4,126,65,150),(5,127,66,141),(6,128,67,142),(7,129,68,143),(8,130,69,144),(9,121,70,145),(10,122,61,146),(11,78,153,100),(12,79,154,91),(13,80,155,92),(14,71,156,93),(15,72,157,94),(16,73,158,95),(17,74,159,96),(18,75,160,97),(19,76,151,98),(20,77,152,99),(21,82,34,106),(22,83,35,107),(23,84,36,108),(24,85,37,109),(25,86,38,110),(26,87,39,101),(27,88,40,102),(28,89,31,103),(29,90,32,104),(30,81,33,105),(41,112,59,134),(42,113,60,135),(43,114,51,136),(44,115,52,137),(45,116,53,138),(46,117,54,139),(47,118,55,140),(48,119,56,131),(49,120,57,132),(50,111,58,133)], [(1,79,56,107),(2,80,57,108),(3,71,58,109),(4,72,59,110),(5,73,60,101),(6,74,51,102),(7,75,52,103),(8,76,53,104),(9,77,54,105),(10,78,55,106),(11,118,34,146),(12,119,35,147),(13,120,36,148),(14,111,37,149),(15,112,38,150),(16,113,39,141),(17,114,40,142),(18,115,31,143),(19,116,32,144),(20,117,33,145),(21,122,153,140),(22,123,154,131),(23,124,155,132),(24,125,156,133),(25,126,157,134),(26,127,158,135),(27,128,159,136),(28,129,160,137),(29,130,151,138),(30,121,152,139),(41,86,65,94),(42,87,66,95),(43,88,67,96),(44,89,68,97),(45,90,69,98),(46,81,70,99),(47,82,61,100),(48,83,62,91),(49,84,63,92),(50,85,64,93)], [(11,153),(12,154),(13,155),(14,156),(15,157),(16,158),(17,159),(18,160),(19,151),(20,152),(21,34),(22,35),(23,36),(24,37),(25,38),(26,39),(27,40),(28,31),(29,32),(30,33),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,149),(112,150),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,139),(122,140),(123,131),(124,132),(125,133),(126,134),(127,135),(128,136),(129,137),(130,138)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4L | 4M | ··· | 4R | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AJ | 20A | ··· | 20AV | 20AW | ··· | 20BT |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C4oD4 | C5xC4oD4 |
| kernel | C10xC42:2C2 | C2xC4xC20 | C10xC22:C4 | C10xC4:C4 | C5xC42:2C2 | C2xC42:2C2 | C2xC42 | C2xC22:C4 | C2xC4:C4 | C42:2C2 | C2xC10 | C22 |
| # reps | 1 | 1 | 3 | 3 | 8 | 4 | 4 | 12 | 12 | 32 | 12 | 48 |
Matrix representation of C10xC42:2C2 ►in GL5(F41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 | 0 |
| 0 | 0 | 23 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 20 | 39 | 0 | 0 |
| 0 | 16 | 21 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 20 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,23,0,0,0,0,0,23,0,0,0,0,0,18,0,0,0,0,0,18],[40,0,0,0,0,0,20,16,0,0,0,39,21,0,0,0,0,0,32,0,0,0,0,0,32],[40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,40,0],[40,0,0,0,0,0,1,20,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40] >;
C10xC42:2C2 in GAP, Magma, Sage, TeX
C_{10}\times C_4^2\rtimes_2C_2 % in TeX
G:=Group("C10xC4^2:2C2"); // GroupNames label
G:=SmallGroup(320,1530);
// by ID
G=gap.SmallGroup(320,1530);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1688,3446,436]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations