direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C42⋊2C2, (C2×C42)⋊4C10, C42⋊14(C2×C10), (C4×C20)⋊49C22, C24.14(C2×C10), (C2×C20).712C23, (C2×C10).348C24, C23.72(C22×C10), C22.22(C23×C10), (C23×C10).14C22, (C22×C10).86C23, (C22×C20).510C22, (C2×C4×C20)⋊6C2, (C10×C4⋊C4)⋊44C2, (C2×C4⋊C4)⋊17C10, C4⋊C4⋊13(C2×C10), (C5×C4⋊C4)⋊69C22, C2.11(C10×C4○D4), C10.230(C2×C4○D4), C22.34(C5×C4○D4), (C2×C22⋊C4).12C10, C22⋊C4.11(C2×C10), (C10×C22⋊C4).32C2, (C2×C4).16(C22×C10), (C2×C10).234(C4○D4), (C22×C4).102(C2×C10), (C5×C22⋊C4).145C22, SmallGroup(320,1530)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 354 in 246 conjugacy classes, 162 normal (12 characteristic)
C1, C2 [×7], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×12], C2×C4 [×12], C23, C23 [×2], C23 [×6], C10 [×7], C10 [×2], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C24, C20 [×12], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C42⋊2C2 [×8], C2×C20 [×12], C2×C20 [×12], C22×C10, C22×C10 [×2], C22×C10 [×6], C2×C42⋊2C2, C4×C20 [×4], C5×C22⋊C4 [×12], C5×C4⋊C4 [×12], C22×C20 [×6], C23×C10, C2×C4×C20, C10×C22⋊C4 [×3], C10×C4⋊C4 [×3], C5×C42⋊2C2 [×8], C10×C42⋊2C2
Quotients:
C1, C2 [×15], C22 [×35], C5, C23 [×15], C10 [×15], C4○D4 [×6], C24, C2×C10 [×35], C42⋊2C2 [×4], C2×C4○D4 [×3], C22×C10 [×15], C2×C42⋊2C2, C5×C4○D4 [×6], C23×C10, C5×C42⋊2C2 [×4], C10×C4○D4 [×3], C10×C42⋊2C2
Generators and relations
G = < a,b,c,d | a10=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >
►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 55 140)(2 124 56 131)(3 125 57 132)(4 126 58 133)(5 127 59 134)(6 128 60 135)(7 129 51 136)(8 130 52 137)(9 121 53 138)(10 122 54 139)(11 89 31 96)(12 90 32 97)(13 81 33 98)(14 82 34 99)(15 83 35 100)(16 84 36 91)(17 85 37 92)(18 86 38 93)(19 87 39 94)(20 88 40 95)(21 73 158 106)(22 74 159 107)(23 75 160 108)(24 76 151 109)(25 77 152 110)(26 78 153 101)(27 79 154 102)(28 80 155 103)(29 71 156 104)(30 72 157 105)(41 112 70 145)(42 113 61 146)(43 114 62 147)(44 115 63 148)(45 116 64 149)(46 117 65 150)(47 118 66 141)(48 119 67 142)(49 120 68 143)(50 111 69 144)
(1 79 67 100)(2 80 68 91)(3 71 69 92)(4 72 70 93)(5 73 61 94)(6 74 62 95)(7 75 63 96)(8 76 64 97)(9 77 65 98)(10 78 66 99)(11 129 160 148)(12 130 151 149)(13 121 152 150)(14 122 153 141)(15 123 154 142)(16 124 155 143)(17 125 156 144)(18 126 157 145)(19 127 158 146)(20 128 159 147)(21 113 39 134)(22 114 40 135)(23 115 31 136)(24 116 32 137)(25 117 33 138)(26 118 34 139)(27 119 35 140)(28 120 36 131)(29 111 37 132)(30 112 38 133)(41 86 58 105)(42 87 59 106)(43 88 60 107)(44 89 51 108)(45 90 52 109)(46 81 53 110)(47 82 54 101)(48 83 55 102)(49 84 56 103)(50 85 57 104)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 158)(22 159)(23 160)(24 151)(25 152)(26 153)(27 154)(28 155)(29 156)(30 157)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 101)(100 102)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 131)(121 150)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)
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), (1,123,55,140)(2,124,56,131)(3,125,57,132)(4,126,58,133)(5,127,59,134)(6,128,60,135)(7,129,51,136)(8,130,52,137)(9,121,53,138)(10,122,54,139)(11,89,31,96)(12,90,32,97)(13,81,33,98)(14,82,34,99)(15,83,35,100)(16,84,36,91)(17,85,37,92)(18,86,38,93)(19,87,39,94)(20,88,40,95)(21,73,158,106)(22,74,159,107)(23,75,160,108)(24,76,151,109)(25,77,152,110)(26,78,153,101)(27,79,154,102)(28,80,155,103)(29,71,156,104)(30,72,157,105)(41,112,70,145)(42,113,61,146)(43,114,62,147)(44,115,63,148)(45,116,64,149)(46,117,65,150)(47,118,66,141)(48,119,67,142)(49,120,68,143)(50,111,69,144), (1,79,67,100)(2,80,68,91)(3,71,69,92)(4,72,70,93)(5,73,61,94)(6,74,62,95)(7,75,63,96)(8,76,64,97)(9,77,65,98)(10,78,66,99)(11,129,160,148)(12,130,151,149)(13,121,152,150)(14,122,153,141)(15,123,154,142)(16,124,155,143)(17,125,156,144)(18,126,157,145)(19,127,158,146)(20,128,159,147)(21,113,39,134)(22,114,40,135)(23,115,31,136)(24,116,32,137)(25,117,33,138)(26,118,34,139)(27,119,35,140)(28,120,36,131)(29,111,37,132)(30,112,38,133)(41,86,58,105)(42,87,59,106)(43,88,60,107)(44,89,51,108)(45,90,52,109)(46,81,53,110)(47,82,54,101)(48,83,55,102)(49,84,56,103)(50,85,57,104), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,158)(22,159)(23,160)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,131)(121,150)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)>;
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), (1,123,55,140)(2,124,56,131)(3,125,57,132)(4,126,58,133)(5,127,59,134)(6,128,60,135)(7,129,51,136)(8,130,52,137)(9,121,53,138)(10,122,54,139)(11,89,31,96)(12,90,32,97)(13,81,33,98)(14,82,34,99)(15,83,35,100)(16,84,36,91)(17,85,37,92)(18,86,38,93)(19,87,39,94)(20,88,40,95)(21,73,158,106)(22,74,159,107)(23,75,160,108)(24,76,151,109)(25,77,152,110)(26,78,153,101)(27,79,154,102)(28,80,155,103)(29,71,156,104)(30,72,157,105)(41,112,70,145)(42,113,61,146)(43,114,62,147)(44,115,63,148)(45,116,64,149)(46,117,65,150)(47,118,66,141)(48,119,67,142)(49,120,68,143)(50,111,69,144), (1,79,67,100)(2,80,68,91)(3,71,69,92)(4,72,70,93)(5,73,61,94)(6,74,62,95)(7,75,63,96)(8,76,64,97)(9,77,65,98)(10,78,66,99)(11,129,160,148)(12,130,151,149)(13,121,152,150)(14,122,153,141)(15,123,154,142)(16,124,155,143)(17,125,156,144)(18,126,157,145)(19,127,158,146)(20,128,159,147)(21,113,39,134)(22,114,40,135)(23,115,31,136)(24,116,32,137)(25,117,33,138)(26,118,34,139)(27,119,35,140)(28,120,36,131)(29,111,37,132)(30,112,38,133)(41,86,58,105)(42,87,59,106)(43,88,60,107)(44,89,51,108)(45,90,52,109)(46,81,53,110)(47,82,54,101)(48,83,55,102)(49,84,56,103)(50,85,57,104), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,158)(22,159)(23,160)(24,151)(25,152)(26,153)(27,154)(28,155)(29,156)(30,157)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,103)(92,104)(93,105)(94,106)(95,107)(96,108)(97,109)(98,110)(99,101)(100,102)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,131)(121,150)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149) );
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)], [(1,123,55,140),(2,124,56,131),(3,125,57,132),(4,126,58,133),(5,127,59,134),(6,128,60,135),(7,129,51,136),(8,130,52,137),(9,121,53,138),(10,122,54,139),(11,89,31,96),(12,90,32,97),(13,81,33,98),(14,82,34,99),(15,83,35,100),(16,84,36,91),(17,85,37,92),(18,86,38,93),(19,87,39,94),(20,88,40,95),(21,73,158,106),(22,74,159,107),(23,75,160,108),(24,76,151,109),(25,77,152,110),(26,78,153,101),(27,79,154,102),(28,80,155,103),(29,71,156,104),(30,72,157,105),(41,112,70,145),(42,113,61,146),(43,114,62,147),(44,115,63,148),(45,116,64,149),(46,117,65,150),(47,118,66,141),(48,119,67,142),(49,120,68,143),(50,111,69,144)], [(1,79,67,100),(2,80,68,91),(3,71,69,92),(4,72,70,93),(5,73,61,94),(6,74,62,95),(7,75,63,96),(8,76,64,97),(9,77,65,98),(10,78,66,99),(11,129,160,148),(12,130,151,149),(13,121,152,150),(14,122,153,141),(15,123,154,142),(16,124,155,143),(17,125,156,144),(18,126,157,145),(19,127,158,146),(20,128,159,147),(21,113,39,134),(22,114,40,135),(23,115,31,136),(24,116,32,137),(25,117,33,138),(26,118,34,139),(27,119,35,140),(28,120,36,131),(29,111,37,132),(30,112,38,133),(41,86,58,105),(42,87,59,106),(43,88,60,107),(44,89,51,108),(45,90,52,109),(46,81,53,110),(47,82,54,101),(48,83,55,102),(49,84,56,103),(50,85,57,104)], [(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,158),(22,159),(23,160),(24,151),(25,152),(26,153),(27,154),(28,155),(29,156),(30,157),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,103),(92,104),(93,105),(94,106),(95,107),(96,108),(97,109),(98,110),(99,101),(100,102),(111,132),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,131),(121,150),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149)])
Matrix representation ►G ⊆ 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] >;
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 | C4○D4 | C5×C4○D4 |
| kernel | C10×C42⋊2C2 | C2×C4×C20 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C42⋊2C2 | C2×C42⋊2C2 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C42⋊2C2 | C2×C10 | C22 |
| # reps | 1 | 1 | 3 | 3 | 8 | 4 | 4 | 12 | 12 | 32 | 12 | 48 |
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
×
⋊
ℤ
𝔽
○
≀