metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D20⋊10C4, C4.63(C2×D20), C4⋊C4.235D10, D20.40(C2×C4), (C2×C4).146D20, C20.143(C2×D4), (C2×C20).496D4, C42⋊C2⋊2D5, D20⋊6C4⋊43C2, C10.85(C4○D8), C10.Q16⋊43C2, (C22×C10).75D4, (C2×C20).329C23, C20.124(C22×C4), Dic10.41(C2×C4), (C22×C4).339D10, C23.38(C5⋊D4), C5⋊4(C23.24D4), C4.52(D10⋊C4), C20.111(C22⋊C4), C2.2(D4.8D10), (C2×D20).243C22, C22.1(D10⋊C4), (C22×C20).151C22, (C2×Dic10).270C22, C4.52(C2×C4×D5), (C2×C4).81(C4×D5), (C22×C5⋊2C8)⋊2C2, (C2×C4○D20).7C2, (C2×C20).265(C2×C4), (C5×C42⋊C2)⋊2C2, (C2×C10).458(C2×D4), C10.86(C2×C22⋊C4), C22.73(C2×C5⋊D4), C2.18(C2×D10⋊C4), (C2×C4).274(C5⋊D4), (C5×C4⋊C4).266C22, (C2×C4).429(C22×D5), (C2×C10).79(C22⋊C4), (C2×C5⋊2C8).251C22, SmallGroup(320,629)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4○D20⋊10C4
G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, ad=da, cbc=a2b9, dbd-1=a2b, dcd-1=a2b5c >
Subgroups: 590 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, C20, D10, C2×C10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C23.24D4, C2×C5⋊2C8, C2×C5⋊2C8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, D20⋊6C4, C10.Q16, C22×C5⋊2C8, C5×C42⋊C2, C2×C4○D20, C4○D20⋊10C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4○D8, C4×D5, D20, C5⋊D4, C22×D5, C23.24D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.8D10, C4○D20⋊10C4
►On 160 points
Generators in S160
(1 94 11 84)(2 95 12 85)(3 96 13 86)(4 97 14 87)(5 98 15 88)(6 99 16 89)(7 100 17 90)(8 81 18 91)(9 82 19 92)(10 83 20 93)(21 102 31 112)(22 103 32 113)(23 104 33 114)(24 105 34 115)(25 106 35 116)(26 107 36 117)(27 108 37 118)(28 109 38 119)(29 110 39 120)(30 111 40 101)(41 122 51 132)(42 123 52 133)(43 124 53 134)(44 125 54 135)(45 126 55 136)(46 127 56 137)(47 128 57 138)(48 129 58 139)(49 130 59 140)(50 131 60 121)(61 154 71 144)(62 155 72 145)(63 156 73 146)(64 157 74 147)(65 158 75 148)(66 159 76 149)(67 160 77 150)(68 141 78 151)(69 142 79 152)(70 143 80 153)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
(1 75 33 52)(2 66 34 43)(3 77 35 54)(4 68 36 45)(5 79 37 56)(6 70 38 47)(7 61 39 58)(8 72 40 49)(9 63 21 60)(10 74 22 51)(11 65 23 42)(12 76 24 53)(13 67 25 44)(14 78 26 55)(15 69 27 46)(16 80 28 57)(17 71 29 48)(18 62 30 59)(19 73 31 50)(20 64 32 41)(81 145 101 130)(82 156 102 121)(83 147 103 132)(84 158 104 123)(85 149 105 134)(86 160 106 125)(87 151 107 136)(88 142 108 127)(89 153 109 138)(90 144 110 129)(91 155 111 140)(92 146 112 131)(93 157 113 122)(94 148 114 133)(95 159 115 124)(96 150 116 135)(97 141 117 126)(98 152 118 137)(99 143 119 128)(100 154 120 139)
G:=sub<Sym(160)| (1,94,11,84)(2,95,12,85)(3,96,13,86)(4,97,14,87)(5,98,15,88)(6,99,16,89)(7,100,17,90)(8,81,18,91)(9,82,19,92)(10,83,20,93)(21,102,31,112)(22,103,32,113)(23,104,33,114)(24,105,34,115)(25,106,35,116)(26,107,36,117)(27,108,37,118)(28,109,38,119)(29,110,39,120)(30,111,40,101)(41,122,51,132)(42,123,52,133)(43,124,53,134)(44,125,54,135)(45,126,55,136)(46,127,56,137)(47,128,57,138)(48,129,58,139)(49,130,59,140)(50,131,60,121)(61,154,71,144)(62,155,72,145)(63,156,73,146)(64,157,74,147)(65,158,75,148)(66,159,76,149)(67,160,77,150)(68,141,78,151)(69,142,79,152)(70,143,80,153), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151), (1,75,33,52)(2,66,34,43)(3,77,35,54)(4,68,36,45)(5,79,37,56)(6,70,38,47)(7,61,39,58)(8,72,40,49)(9,63,21,60)(10,74,22,51)(11,65,23,42)(12,76,24,53)(13,67,25,44)(14,78,26,55)(15,69,27,46)(16,80,28,57)(17,71,29,48)(18,62,30,59)(19,73,31,50)(20,64,32,41)(81,145,101,130)(82,156,102,121)(83,147,103,132)(84,158,104,123)(85,149,105,134)(86,160,106,125)(87,151,107,136)(88,142,108,127)(89,153,109,138)(90,144,110,129)(91,155,111,140)(92,146,112,131)(93,157,113,122)(94,148,114,133)(95,159,115,124)(96,150,116,135)(97,141,117,126)(98,152,118,137)(99,143,119,128)(100,154,120,139)>;
G:=Group( (1,94,11,84)(2,95,12,85)(3,96,13,86)(4,97,14,87)(5,98,15,88)(6,99,16,89)(7,100,17,90)(8,81,18,91)(9,82,19,92)(10,83,20,93)(21,102,31,112)(22,103,32,113)(23,104,33,114)(24,105,34,115)(25,106,35,116)(26,107,36,117)(27,108,37,118)(28,109,38,119)(29,110,39,120)(30,111,40,101)(41,122,51,132)(42,123,52,133)(43,124,53,134)(44,125,54,135)(45,126,55,136)(46,127,56,137)(47,128,57,138)(48,129,58,139)(49,130,59,140)(50,131,60,121)(61,154,71,144)(62,155,72,145)(63,156,73,146)(64,157,74,147)(65,158,75,148)(66,159,76,149)(67,160,77,150)(68,141,78,151)(69,142,79,152)(70,143,80,153), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,24)(22,23)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,47)(42,46)(43,45)(48,60)(49,59)(50,58)(51,57)(52,56)(53,55)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,86)(82,85)(83,84)(87,100)(88,99)(89,98)(90,97)(91,96)(92,95)(93,94)(101,106)(102,105)(103,104)(107,120)(108,119)(109,118)(110,117)(111,116)(112,115)(113,114)(121,129)(122,128)(123,127)(124,126)(130,140)(131,139)(132,138)(133,137)(134,136)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151), (1,75,33,52)(2,66,34,43)(3,77,35,54)(4,68,36,45)(5,79,37,56)(6,70,38,47)(7,61,39,58)(8,72,40,49)(9,63,21,60)(10,74,22,51)(11,65,23,42)(12,76,24,53)(13,67,25,44)(14,78,26,55)(15,69,27,46)(16,80,28,57)(17,71,29,48)(18,62,30,59)(19,73,31,50)(20,64,32,41)(81,145,101,130)(82,156,102,121)(83,147,103,132)(84,158,104,123)(85,149,105,134)(86,160,106,125)(87,151,107,136)(88,142,108,127)(89,153,109,138)(90,144,110,129)(91,155,111,140)(92,146,112,131)(93,157,113,122)(94,148,114,133)(95,159,115,124)(96,150,116,135)(97,141,117,126)(98,152,118,137)(99,143,119,128)(100,154,120,139) );
G=PermutationGroup([[(1,94,11,84),(2,95,12,85),(3,96,13,86),(4,97,14,87),(5,98,15,88),(6,99,16,89),(7,100,17,90),(8,81,18,91),(9,82,19,92),(10,83,20,93),(21,102,31,112),(22,103,32,113),(23,104,33,114),(24,105,34,115),(25,106,35,116),(26,107,36,117),(27,108,37,118),(28,109,38,119),(29,110,39,120),(30,111,40,101),(41,122,51,132),(42,123,52,133),(43,124,53,134),(44,125,54,135),(45,126,55,136),(46,127,56,137),(47,128,57,138),(48,129,58,139),(49,130,59,140),(50,131,60,121),(61,154,71,144),(62,155,72,145),(63,156,73,146),(64,157,74,147),(65,158,75,148),(66,159,76,149),(67,160,77,150),(68,141,78,151),(69,142,79,152),(70,143,80,153)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,24),(22,23),(25,40),(26,39),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,47),(42,46),(43,45),(48,60),(49,59),(50,58),(51,57),(52,56),(53,55),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,86),(82,85),(83,84),(87,100),(88,99),(89,98),(90,97),(91,96),(92,95),(93,94),(101,106),(102,105),(103,104),(107,120),(108,119),(109,118),(110,117),(111,116),(112,115),(113,114),(121,129),(122,128),(123,127),(124,126),(130,140),(131,139),(132,138),(133,137),(134,136),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)], [(1,75,33,52),(2,66,34,43),(3,77,35,54),(4,68,36,45),(5,79,37,56),(6,70,38,47),(7,61,39,58),(8,72,40,49),(9,63,21,60),(10,74,22,51),(11,65,23,42),(12,76,24,53),(13,67,25,44),(14,78,26,55),(15,69,27,46),(16,80,28,57),(17,71,29,48),(18,62,30,59),(19,73,31,50),(20,64,32,41),(81,145,101,130),(82,156,102,121),(83,147,103,132),(84,158,104,123),(85,149,105,134),(86,160,106,125),(87,151,107,136),(88,142,108,127),(89,153,109,138),(90,144,110,129),(91,155,111,140),(92,146,112,131),(93,157,113,122),(94,148,114,133),(95,159,115,124),(96,150,116,135),(97,141,117,126),(98,152,118,137),(99,143,119,128),(100,154,120,139)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | D4.8D10 |
| kernel | C4○D20⋊10C4 | D20⋊6C4 | C10.Q16 | C22×C5⋊2C8 | C5×C42⋊C2 | C2×C4○D20 | C4○D20 | C2×C20 | C22×C10 | C42⋊C2 | C4⋊C4 | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 8 | 4 | 4 | 8 |
Matrix representation of C4○D20⋊10C4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 35 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 18 |
| 0 | 0 | 9 | 1 |
| 40 | 35 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 18 |
| 0 | 0 | 0 | 1 |
| 39 | 13 | 0 | 0 |
| 28 | 2 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 26 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[35,1,0,0,40,0,0,0,0,0,40,9,0,0,18,1],[40,0,0,0,35,1,0,0,0,0,40,0,0,0,18,1],[39,28,0,0,13,2,0,0,0,0,0,26,0,0,30,0] >;
C4○D20⋊10C4 in GAP, Magma, Sage, TeX
C_4\circ D_{20}\rtimes_{10}C_4 % in TeX
G:=Group("C4oD20:10C4"); // GroupNames label
G:=SmallGroup(320,629);
// by ID
G=gap.SmallGroup(320,629);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=a^2*b^9,d*b*d^-1=a^2*b,d*c*d^-1=a^2*b^5*c>;
// generators/relations